text:
Copied to
clipboard

## G = S3×C22×C12order 288 = 25·32

### Direct product of C22×C12 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C22×C12
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C22×C6 — S3×C22×C12
 Lower central C3 — S3×C22×C12
 Upper central C1 — C22×C12

Generators and relations for S3×C22×C12
G = < a,b,c,d,e | a2=b2=c12=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 890 in 499 conjugacy classes, 290 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×2], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×28], S3 [×8], C6 [×2], C6 [×12], C6 [×15], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], C32, Dic3 [×4], C12 [×8], C12 [×8], D6 [×28], C2×C6 [×14], C2×C6 [×35], C22×C4, C22×C4 [×13], C24, C3×S3 [×8], C3×C6, C3×C6 [×6], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×12], C2×C12 [×28], C22×S3 [×14], C22×C6 [×2], C22×C6 [×15], C23×C4, C3×Dic3 [×4], C3×C12 [×4], S3×C6 [×28], C62 [×7], S3×C2×C4 [×12], C22×Dic3, C22×C12 [×2], C22×C12 [×14], S3×C23, C23×C6, S3×C12 [×16], C6×Dic3 [×6], C6×C12 [×6], S3×C2×C6 [×14], C2×C62, S3×C22×C4, C23×C12, S3×C2×C12 [×12], Dic3×C2×C6, C2×C6×C12, S3×C22×C6, S3×C22×C12
Quotients: C1, C2 [×15], C3, C4 [×8], C22 [×35], S3, C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], D6 [×7], C2×C6 [×35], C22×C4 [×14], C24, C3×S3, C4×S3 [×4], C2×C12 [×28], C22×S3 [×7], C22×C6 [×15], C23×C4, S3×C6 [×7], S3×C2×C4 [×6], C22×C12 [×14], S3×C23, C23×C6, S3×C12 [×4], S3×C2×C6 [×7], S3×C22×C4, C23×C12, S3×C2×C12 [×6], S3×C22×C6, S3×C22×C12

Smallest permutation representation of S3×C22×C12
On 96 points
Generators in S96
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(49 82)(50 83)(51 84)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 94)(74 95)(75 96)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 49)(11 50)(12 51)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 85)(47 86)(48 87)

G:=sub<Sym(96)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(49,82)(50,83)(51,84)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87)>;

G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(49,82)(50,83)(51,84)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87) );

G=PermutationGroup([(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,46),(14,47),(15,48),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(49,82),(50,83),(51,84),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,94),(74,95),(75,96),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91),(83,92),(84,93)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,49),(11,50),(12,51),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,85),(47,86),(48,87)])

144 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 3E 4A ··· 4H 4I ··· 4P 6A ··· 6N 6O ··· 6AI 6AJ ··· 6AY 12A ··· 12P 12Q ··· 12AN 12AO ··· 12BD order 1 2 ··· 2 2 ··· 2 3 3 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 ··· 1 3 ··· 3 1 1 2 2 2 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 D6 D6 C3×S3 C4×S3 S3×C6 S3×C6 S3×C12 kernel S3×C22×C12 S3×C2×C12 Dic3×C2×C6 C2×C6×C12 S3×C22×C6 S3×C22×C4 S3×C2×C6 S3×C2×C4 C22×Dic3 C22×C12 S3×C23 C22×S3 C22×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 1 1 2 16 24 2 2 2 32 1 6 1 2 8 12 2 16

Matrix representation of S3×C22×C12 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 11 0 0 0 0 4 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 1 9
,
 1 0 0 0 0 1 0 0 0 0 10 8 0 0 12 3
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,4,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,3,1,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,10,12,0,0,8,3] >;

S3×C22×C12 in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_{12}
% in TeX

G:=Group("S3xC2^2xC12");
// GroupNames label

G:=SmallGroup(288,989);
// by ID

G=gap.SmallGroup(288,989);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,192,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^12=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽


text:
# What is the loudest possible sound?

For a long time, Wikipedia has said that the loudest possible sound is 191 dB SPL, as this corresponds to 1 atmosphere of pressure peak-to-peak, and anything above this would be clipped at vacuum on the negative peaks, and is therefore classed as a "shockwave" rather than "sound". (Though Wikipedia also defines a shock wave as a wave moving faster than the speed of sound, regardless of amplitude.)

It gives no references, however, and I've since learned that pressure waves in air are always non-linear, and the science of acoustics assumes linearity and small pressure levels to simplify calculations. So the air will already be distorted before this pressure level.

So is there a commonly-held definition of when distortion becomes too great to consider a wave "sound"? At what dB SPL is it? Is it possible to calculate the amplitude that a sine wave in air would be distorted by 1% THD, for instance?

The "commonly held definition" is the wikipedia one... it's not so much a question of distortion, as a question that a wave is symmetrical - that is, it should not result in a net motion of gas. It is possible to construct a sinusoidal pressure wave with a peak pressure of 2 atm and a valley of 0 - from a displacement perspective this is a distorted wave, but from a pressure perspective it is not.

However, you cannot do that if the peak pressure is greater than 2 atmosphere.

• What do you mean by "from a displacement perspective"? – endolith Jan 6 '17 at 21:01
• @endolith If the sound frequency is high enough to consider the motion of the gas is adiabatic, then $PV^\gamma = \text{constant}$ where $\gamma = 1.4$ for air, so the velocity (and hence the displacement) is not a linear function of the pressure. For "normal amplitude" sound waves there is only a small change in $P$ about its mean value. Of course for a low frequency, high amplitude sound wave, the adiabatic assumption may not be strictly valid because of heat conduction in the gas. – alephzero Jan 6 '17 at 23:52
• Is it possible to calculate distortion percentage of displacement wave for a given SPL? – endolith Jan 7 '17 at 0:28

format: plain


text:
# Shortening time scale to reduce thermal effects in quantum transistors

## Abstract

In this article, we present a quantum transistor model based on a network of coupled quantum oscillators destined to quantum information processing tasks in linear optics. To this end, we show in an analytical way how a set of N quantum oscillators (data-bus) can be used as an optical quantum switch, in which the energy gap of the data bus oscillators plays the role of an adjustable “potential barrier”. This enables us to “block or allow” the quantum information to flow from the source to the drain. In addition, we discuss how this device can be useful for implementing single qubit phase-shift quantum gates with high fidelity, so that it can be used as a useful tool. To conclude, during the study of the performance of our device when considering the interaction of this with a thermal reservoir, we highlight the important role played by the set of oscillators which constitute the data-bus in reducing the unwanted effects of the thermal reservoir. This is achieved by reducing the information exchange time (shortening time scale) between the desired oscillators. In particular, we have identified a non-trivial criterion in which the ideal size of the data-bus can be obtained so that it presents the best possible performance. We believe that our study can be perfectly adapted to a large number of thermal reservoir models.

## Introduction

Providing small devices that operate in quantum regime, maintaining high processing fidelity against the effects of decoherence, is of great importance in both quantum computing and information processing. Motivated by the important role played by transistors in classical devices, we can imagine how the quantum version of such electronic components could help us to achieve even more significant advances in quantum information processing. In classical devices, a transistor can be used as a switch to block or transfer classical information (encoded in terms of the intensity of electric current, for example) from some source to a drain. On the other hand, differently from the classical transistor, because of the non-clone theorem1, a quantum transistor can not copy arbitrary quantum information encoded in the source. Therefore, by making an analogy with its classical counterpart, quantum transistors could be used to block or allow the flux of quantum information from a source to a drain. So that, to design a quantum transistor, we must focus on the performance of the quantum switch, trying to make it as efficient as possible. In this scenario, we need to study the transfer of quantum information between two quantum systems (quantum bits - qubits). A greater motivation to design a quantum transistor is associated with its applicability in quantum computation, as shown in the context of adiabatic quantum computation2, where fault-tolerant universal quantum computation can be efficiently achieved if we can build an “adiabatic quantum transistor”. Besides that, the quantum transistors were studied in the adiabatic quantum computing scenario2,3, spin chain4,5, ultra-cold atoms6,7,8, and in the other systems presented in many refs. 9,10,11,12,13,14. As a contribution of this paper, we discuss how a bosonic quantum transistor could be designed by using a particular arrangement of coupled quantum harmonic oscillators, providing therefore a quantum device for blocking and/or transferring quantum information in linear optics.

In the literature there are similar works to what we aim to investigate here under the approach of Quantum State Transfer (QST)15,16,17. In the works15,16, the authors investigate some particular networks oscillators in the strong coupling regime, in which they verify that the transfer of some particular coherent states occur in a short time scale inversely proportional to the square root of the number of oscillators, as we verify here. Although the authors perceive that the effects of decoherence on this time scale are reduced, an analysis of the temperature effects is absent. In ref.17 the authors analyze the QST in a linear chain of N constituents from the perspective of an adiabatic dynamics. In this work the authors also manage to inhibit the effects of decoherence to the situation in which the state of the reservoir is the vacuum. They show that the fidelity is so close to unity the smaller the ratio between the decay rate γ and the coupling intensity between the constituents for a linear network with N = 39 elements. In order to extend some of the existing works in the literature, we intend to investigate how the size of the data bus in a specific (and different) arrangement can be useful to inhibit the effects of decoherence in the presence of a thermal reservoir.

In this paper, we present a quantum transistor model that can be useful for quantum information processing in linear optics. To this end, we consider that two quantum oscillators (source and drain) are coupled to each other only indirectly through one or even a network composed of N non-interacting quantum oscillators (data-bus) which play the role of the transistor gate (the quantum switch). Through the study carried out on this system, it is possible to demonstrate that the performance of quantum information blocking of our transistor is associated with the detuning between the resonant frequencies of the source and drain oscillators with the frequencies of the data-bus oscillators. By using the transistor we propose in this work, which is genuinely quantum, one show that when the gate is opened to transfer quantum information, we can adjust many parameters (frequencies, coupling strengths and the number of data-bus oscillators), or just a few, in order to implement simple qubit logical quantum gates associated to phase shift gates. In this sense, our study provides a model that makes logical quantum gates from quantum transistors, as an alternative to adiabatic quantum transistors2. Finally, we study the performance against the decoherence of the quantum transistor model.

## Results

Any new quantum transistor proposal must be composed of three fundamental parts: source, gate and drain. If we want to use it in quantum computing, the most appropriate way would be to consider the source and drain as two-level systems (a single qubit), whereas the gate represents a quantum channel (consisting of one or more qubits) that indirectly connects the source with the drain.

Let us consider hereafter that the subscript $$\ell =s,d$$, where s (d) represents the source (drain) oscillator, and the gate oscillators are labeled by m and/or n. In general, considering a gate as a network of non-interacting quantum oscillators, instead of a single oscillator, the Hamiltonian that describes the dynamics of this system can be written as H = H0 + V, in which we defined

$${H}_{0}=\hslash \sum _{\ell }{\omega }_{\ell }{a}_{\ell }^{\dagger }{a}_{\ell }+\hslash \sum _{m}{\omega }_{m}{a}_{m}^{\dagger }{a}_{m},\,{\rm{and}}\,V=\hslash \sum _{m,\ell }({\lambda }_{\ell m}{a}_{\ell }^{\dagger }{a}_{m}+{\lambda }_{\ell m}^{\ast }{a}_{\ell }{a}_{m}^{\dagger }),$$
(1)

such that ω’s are the natural frequencies of respective oscillators, whereas λ’s are the coupling strengths between two oscillators characterized by the pair of subscripted indices. The operators a’s (a’s) represent the creation (annihilation) of a quanta in the respective oscillator. In left Fig. 1 we consider a transistor composed of three parts, each part consisting of a quantum oscillator: the source oscillator (the left), the drain oscillator (the right), and the gate oscillator (in the middle) which is coupled to the first two.

### The quantum transistor

Since a quantum transistor must be able to control the quantum information flow, we will consider, in our study, that the state $$|{\psi }_{s}\rangle$$, of the source oscillator, has the information that can be encoded in a quantum bit whereas the state of the drain oscillator and that of the gate are in the vacuum state. In this way, the initial state of the whole system can be written as a tensor product of the states of each oscillator in the form $$|{\rm{\Psi }}(0)\rangle =|{\psi }_{s}\rangle \otimes |{\mathrm{\{0}}_{g}\}\rangle \otimes |{0}_{d}\rangle \equiv |{\psi }_{s},\{{0}_{g}\},{0}_{d}\rangle$$. It is known in the literature that for a non-zero weak coupling between the resonant oscillators that unknown information will flow from the source to the drain in a transfer time τtrans proportional to the coupling18,19,20. The challenge of building a quantum transistor can be achieved when we use a certain system parameter to allow or block this transfer. If we could easily connect and disconnect the gate couplings with the source and drain oscillators, this task would be trivially executed in the situation where all oscillators are resonant. However, in this work we are interested in nontrivial situations, in which the couplings between oscillators are kept constant. To this end, we aim for our quantum transistor to use our ability to increase or decrease the frequency of one or more gate oscillators – thus modifying the interval between the energy levels of this oscillator – so as to simulate the gate in our device.

In order to discuss the behavior of our transistor for various situations of interest, we will restrict ourselves to the parameter settings in Eq. (1) for the situation where the drain and source have same natural frequencies ωs = ωd = ω and the real coupling strengths between the oscillators are identical {λsm} = {λgm} = λ. As the main element of our system, the data bus configuration develops an important role in our transistor, as we shall see below. To illustrate the importance of this component, we will consider that κ data-bus oscillators are in resonance with the source and the drain, meanwhile the others (N − κ) are out of resonance, such that we can write ωm = ω, if 1 ≤ m ≤ κ, otherwise, $${\omega }_{m}=\tilde{\omega }=\omega +{\rm{\Delta }}$$, as shown in right Fig. 1. Starting from the initial arbitrary state of the source oscillator $$|{\psi }_{s}\rangle =\alpha |{0}_{s}\rangle +\beta {e}^{i\theta }|{1}_{s}\rangle$$ and assuming that the time evolution can be performed by the operator $$U(t)={e}^{-iHt/\hslash }$$, then we can conclude that the time evolution of the initial state, using the fact that $$H|{0}_{s},\{{0}_{g}\},{0}_{d}\rangle =0$$, will be given by

$$|{\rm{\Psi }}(t)\rangle =\alpha |{0}_{s},\{{0}_{g}\},{0}_{d}\rangle +\beta {e}^{i\theta }U(t)|{1}_{s},\{{0}_{g}\},{0}_{d}\rangle .$$
(2)

From Eq. (2) it is easy to show that the probabilities ps(t) and pd(t) of finding the original information in the qubit source and drain are, respectively, given by

$${p}_{s}(t)={|\langle {\rm{\Psi }}(0)|{\rm{\Psi }}(t)\rangle |}^{2}={|{\alpha }^{2}+{\beta }^{2}\langle {1}_{s},\{{0}_{g}\},{0}_{d}|U(t){\mathrm{|1}}_{s},\{{0}_{g}\},{0}_{d}\rangle |}^{2},$$
(3)
$${p}_{d}(t)={|\langle {\rm{\Phi }}(0)|{\rm{\Psi }}(t)\rangle |}^{2}={|{\alpha }^{2}+{\beta }^{2}\langle {0}_{s},\{{0}_{g}\},{1}_{d}|U(t){\mathrm{|1}}_{s},\{{0}_{g}\},{0}_{d}\rangle |}^{2},$$
(4)

where we define the state $$|{\rm{\Phi }}(0)\rangle =|{0}_{s},\{{0}_{g}\},{\psi }_{d}\rangle ={\mathrm{|0}}_{s},\{{0}_{g}\}\rangle \otimes (\alpha {\mathrm{|0}}_{d}\rangle +\beta {e}^{i\theta }{\mathrm{|1}}_{d}\rangle )$$.

#### Blocking and transferring quantum information

From Eq. (3) it is possible to show that ps(t) = 1 and pd(t) = |α|4 if, and only if, the matrix element defined by u+(t) = 〈1s, {0g}, 0d|U(t)|1s, {0g}, 0d〉 = 1 and, simultaneously, u(t) = 〈0s, {0g}, 1d|U(t)|1s, {0g}, 0d〉 = 0. For the parameter regime we are considering, it is possible to perform the analytical calculation to determine both eigenvalues and eigenvectors in order to obtain the expressions (see methods section for more details) u±(t) = [Λ(t) ± 1]eiωt/2, where we defined

$${\rm{\Lambda }}(t)={{\mathscr{A}}}_{0}{e}^{-i{R}_{0}t/3}+{{\mathscr{A}}}_{+}{e}^{-i{R}_{+}t/3}+{{\mathscr{A}}}_{-}{e}^{-i{R}_{-}t/3},$$
(5)

the amplitudes (for {j = 0, +, −})

$${{\mathscr{A}}}_{j}={[1+2\kappa {(\frac{3\lambda }{{R}_{j}})}^{2}+2(N-\kappa ){(\frac{3\lambda }{{R}_{j}-{\rm{\Delta }}})}^{2}]}^{-1},$$

the coefficients $${R}_{0}={\rm{\Delta }}+2\sqrt{{{\rm{\Delta }}}^{2}+6N{\lambda }^{2}}\,\cos \,\theta$$ and $${R}_{\pm }={\rm{\Delta }}-\sqrt{{{\rm{\Delta }}}^{2}+6N{\lambda }^{2}}(\cos \,\theta \pm \sqrt{3}\,\sin \,\theta )$$, and the angle θ by

$$\theta =\frac{1}{3}\,\arctan (\sqrt{\frac{{({{\rm{\Delta }}}^{2}+6N{\lambda }^{2})}^{3}}{{{\rm{\Delta }}}^{2}{[{{\rm{\Delta }}}^{2}+9(N-3\kappa ){\lambda }^{2}]}^{2}}-1}).$$

In this way, if we consider the case where $$\lambda /{\rm{\Delta }}\ll {(3\sqrt{N})}^{-1}$$, disregarding quadratic or superior terms, we can approximate $$\sin \,\theta \approx \theta \approx \sqrt{6\kappa }\lambda /{\rm{\Delta }}$$ and cosθ ≈ 1, and, consequently, the frequencies can be approximate to the values R0 ≈ 3Δ, $${R}_{\pm }\approx \mp 3\sqrt{2\kappa }\lambda$$ and the amplitudes to $${{\mathscr{A}}}_{0}\approx 0$$ and $${{\mathscr{A}}}_{\pm }\approx \mathrm{1/4}$$. With these values in hand, one can easily determine the quantities

$${u}_{+}(t)={e}^{-i\omega t}\,{\cos }^{2}(\sqrt{\frac{\kappa }{2}}\lambda t),\,{\rm{a}}{\rm{n}}{\rm{d}}\,{u}_{-}(t)=-{e}^{-i\omega t}\,{\sin }^{2}(\sqrt{\frac{\kappa }{2}}\lambda t).$$
(6)

Note from Eq. (6) that when none of the data bus oscillators is resonant with the source and drain (i.e., κ = 0), we get u + (t) = eiωt and u(t) = 0. This result shows that in the regime where $$\lambda /{\rm{\Delta }}\ll {(3\sqrt{N})}^{-1}$$, the system dynamics becomes the same as a single isolated oscillator evolving over time. The role of this oscillating phase in Eq. (6) can be better understood when we write the evolution of the state $$|{\rm{\Psi }}(0)\rangle$$ in the regime $$\lambda /{\rm{\Delta }}\ll {(3\sqrt{N})}^{-1}$$ with κ = 0, that turns to be

$$|{\rm{\Psi }}(t)\rangle \approx (\alpha |{0}_{s}\rangle +\beta {e}^{i(\theta -\omega t)}|{1}_{s}\rangle )\otimes |\{{0}_{g}\},{0}_{d}\rangle .$$
(7)

Note that the information is maintained at the source, despite the appearance of a time-dependent local phase. As the value of this phase varies, we may have (1 − 2β2)2 ≤ ps(t) ≤ 1, which does not mean that the information flows from the source to the drain (or be partially destroyed), once the drain state remains in the vacuum, as we can see from u(t) = 0. Moreover, it is important to highlight the fact that pd(t) = |α|4 and not zero. This value is not null because the information contained in the state $$|{\rm{\Psi }}(0)\rangle$$ has a component $$|0\rangle$$, whose probability amplitude is $$\alpha =\langle 0|{\rm{\Psi }}(0)\rangle$$, regardless of whether or not there is a dynamic between the source and drain oscillators.

Despite this, the performance of our model is not affected by this “unwanted” phase. In addition, if we let the system evolve indefinitely, whenever time t is a positive integer, n, multiple of the recurrence time τR = 2π/ω, we get exactly the input state $$|{\psi }_{s}\rangle$$ encoded in the source qubit. Further on, we will see that this “unwanted” phase becomes indispensable if we are to use our device to implement quantum gates. Therefore, using the scheme in Fig. 1, it is possible to use the Δ-dissonance to block the quantum information indefinitely. Since the blocking situation is associated with our ability to adjust $$\lambda /{\rm{\Delta }}\ll 1/(3\sqrt{N})$$, our model can be efficiently implemented using quantum dot-cavity systems21,22, coupled-cavity array23,24,25, bosonic lattice systems26,27 or cold atoms28, for example.

To analyze the quantum information transfer it is desirable to imagine that the gate configuration of the transistor (open or closed) should be controlled by the adjustment of a single physical parameter, otherwise we may have some technical difficulties in handling a set of parameters. In this way, as we use the dissonance Δ to close the gate, we need to show how this same parameter could be used to open it. In other words, from Eq. (6) we must discuss how a new adjustment of Δ allows us to obtain pd(t) = 1. If we want that the state $$|{\rm{\Psi }}(0)\rangle$$ will be transferred to the drain oscillator, two adjustments must be made simultaneously: eiωt = −1 and $${\sin }^{2}(\sqrt{\kappa /2}\lambda t)=1$$. These adjustments imply that ωt and $$\sqrt{\kappa \mathrm{/2}}\lambda t$$ must be an odd number multiple of π and π/2, respectively. That is, we need to have t = (2j + 1)π/ω and $$t=(2j{\rm{^{\prime} }}+1)\pi /(\lambda \sqrt{2\kappa })$$, respectively, with j and j′ integers. To find the exact value of the transfer time τtrans, we must find the integers j and j′ that satisfy the equality

$${\tau }_{{\rm{trans}}}=\frac{2j+1}{\omega }\pi =\frac{2j^{\prime} +1}{\lambda \sqrt{2\kappa }}\pi .$$
(8)

Note that this equality can only be satisfied when $$\lambda \sqrt{2\kappa }/\omega$$ is the ratio between two odd numbers C1/C2, which can always be achieved, regardless of the values of λ and ω, with the convenient adjust of κ. Once we have made this adjustment, the transfer time becomes

$${\tau }_{{\rm{t}}{\rm{r}}{\rm{a}}{\rm{n}}{\rm{s}}}=\frac{2j+1}{\omega }\pi \,{\rm{o}}{\rm{r}}\,{\tau }_{{\rm{t}}{\rm{r}}{\rm{a}}{\rm{n}}{\rm{s}}}=\frac{2{j}^{{\rm{^{\prime} }}}+1}{\lambda \sqrt{2\kappa }}\pi ,$$

where we must choose the smallest values of j or j′ for which C1(2j + 1) = C2(2j′ + 1).

To better understand this adjustment of κ, let us consider a numerical example. For the case where ω = 1010 Hz and λ = 104 Hz, we can adjust κ = 211 = 2048 in order to eliminate the powers of two from the decomposition in prime numbers of λ and ω and so $$\lambda \sqrt{2\kappa }/\omega$$ becomes the ratio between two odd numbers. For this choice, we obtain C1 = 1 and C2 = 56 and consequently we have j′ = 0 and j = (106 − 1)/2 = 7812. With these values, the transfer time will be multiple integers of τtrans = π/64λ. Another important point to highlight is the following: since eiωt is a function that oscillates very quickly when compared to $${\sin }^{2}(\sqrt{\kappa /2}\lambda t)$$, we observe the existence of secondary peaks approaching the unit at time $${t}_{{\rm{e}}{\rm{x}}}=\pi /(\lambda \sqrt{2\kappa })$$, such that $${\sin }^{2}(\sqrt{\kappa /2}\lambda {t}_{{\rm{e}}{\rm{x}}})=1$$. This characteristic time of the system, which represents the information exchange time between the source and drain oscillator, is inversely proportional to $$\lambda \sqrt{2\kappa }$$, so that the larger κ, the shorter tex. (In the literature the characteristic time of the system is sometimes called a short time scale, i.e., it is the minimum time required for there to be a significant change in the state due to the dynamics of the system. When we focus on the interaction picture, this time is proportional to the inverse of the coupling and when we look at a network system interacting, this time also becomes to be proportional to the square root of the number of oscillators N15,16). This reduction of time will be the key point to the study of the performance of our device against the effects of the thermal reservoirs, as we will see later.

#### Detuning control with atom-field interaction

In order to obtain a optimum control of our transistor it is necessary that we are able to adjust the detuning parameter Δ between the data bus oscillators frequencies with the source and drain oscillators. This control can be accomplished through a dispersive interaction29 between an atom and the field inside the cavity, for example.

To illustrate this procedure, we consider the Hamiltonian Hdisp = Hfield + Hatom + Hatom−field, where $${H}_{{\rm{field}}}=\hslash {\omega }_{0}{a}_{\ell }^{\dagger }{a}_{\ell }$$ is the free Hamiltonian of a single interacting mode $$\ell$$, $${H}_{{\rm{atom}}}=\hslash n{\sigma }_{z}$$ is the Hamiltonian of a two-level atom and $${H}_{{\rm{atom}}-{\rm{field}}}=\hslash \chi {a}_{\ell }^{\dagger }{a}_{\ell }\,{\sigma }_{3}$$ is the Hamiltonian of the dispersive interaction between the atom and the field, where σz = |e〉〈e| − |g〉〈g| and $${\sigma }_{3}=|i\rangle \langle i|-|e\rangle \langle e|,$$ $$|e\rangle$$ ($$|i\rangle$$) denotes the excited (virtual intermediate) state of the atom. The constant χ = g2/δ is given in terms of the atom-field coupling intensity, g, and the detuning δ = ω0 − ν between the field and atom frequencies. It is important to remember that the validity of this dispersive Hamiltonian is confined in situations where $${g}^{2}n\ll {\delta }^{2}+{\gamma }^{2}$$, where n is the mean number of photons in the field and γ is the spontaneous emission rate. The time evolution of an atom-field state, according to the Hamiltonian Hdisp, will be given by the operator $${U}_{{\rm{disp}}}(t)={e}^{-i{H}_{{\rm{disp}}}t/\hslash }$$. Since the commutator $$[{a}_{\ell }^{\dagger }{a}_{\ell }({\omega }_{0}+\chi {\sigma }_{3}),{\sigma }_{z}]=0$$, we can decompose the time evolution between Hfield + Hatom−field and Hatom, so that for an initial state of the atom-field system given by $$|{\psi }_{{\rm{atom}}-{\rm{field}}}(0)\rangle =(a|0\rangle +b|1\rangle )\otimes |e\rangle$$, we obtain the following state evolved in time

$$\begin{array}{ccc}|{\psi }_{{\rm{a}}{\rm{t}}{\rm{o}}{\rm{m}}-{\rm{f}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}}(t)\rangle & = & {e}^{-\frac{i}{\hslash }{H}_{{\rm{f}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}}t}{e}^{-\frac{i}{\hslash }({H}_{{\rm{f}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}}+{H}_{{\rm{a}}{\rm{t}}{\rm{o}}{\rm{m}}-{\rm{f}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}})t}|e\rangle \otimes (a|0\rangle +b|1\rangle )\\ & = & {e}^{-i({\omega }_{0}-\chi ){a}_{\ell }^{\dagger }{a}_{\ell }}(a|0\rangle +b|1\rangle )\otimes ({e}^{-\frac{i}{\hslash }{H}_{{\rm{f}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}}t}|e\rangle ).\end{array}$$
(9)

Thus, one can see from (9) that the field state is factorized and can therefore be discarded at the end of the process. Moreover, we conclude that the temporal evolution of the field can be determined by the effective Hamiltonian

$${\tilde{H}}_{{\rm{field}}}=\hslash ({\omega }_{0}-\chi ){a}_{\ell }^{\dagger }{a}_{\ell },$$

demonstrating that the field behaves effectively with a shift in the energy ω0 → ω0 − χ, when it interacts dispersively with the atom. The same result can be verified when we take into account the interaction between the quantum oscillators.

### Application to quantum computation

In general, the conditions previously discussed (for transferring and blocking quantum information) lead us to think about what happens if we ignore them. As we will show in this section, by violating the condition ωt = (2j + 1)π (for j = 0, 1, 2, …), but maintaining the condition $$\lambda t=(2j^{\prime} +1)\pi /\sqrt{2\kappa }$$ (for j′ = 0, 1, 2, …), we can implement quantum phase-shift gates. In particular, we are interested in a situation where we simultaneously transfer the information and apply a quantum gate, such that we will define κ > 0 hereafter.

In order to demonstrate how the quantum transistor we propose in this paper allows us to implement a particular set of quantum gates, let us consider the system input state as |ψs,{0g},0d〉. We know that at time $${t}_{{\rm{e}}{\rm{x}}}=\pi /(\lambda \sqrt{2\kappa })$$, the system output state is given by

$$|{\rm{\Psi }}({t}_{{\rm{e}}{\rm{x}}})\rangle =|{0}_{s},\{{0}_{g}\}\rangle \otimes (\alpha |{0}_{d}\rangle -\beta {e}^{i[\theta -\omega \pi /(\lambda \sqrt{2\kappa })]}|{1}_{d}\rangle ).$$
(10)

From Eq. (10), it can be seen that the output state is identical to the input state, except for a local phase that must be applied to the state |1〉. This result resembles that obtained by the phase shift gates R(ϕ), which are single-qubit gates that can be combined with other one- and/or two-qubit gates to provide a set of universal quantum gates30,31. In general, given any input state $$|{\psi }_{{\rm{inp}}}\rangle =\alpha |0\rangle +\beta {e}^{i\theta }|1\rangle$$, after the unitary operation R(ϕ), the output state becomes $$|{\psi }_{{\rm{out}}}(\varphi )\rangle =R(\varphi )|{\psi }_{{\rm{inp}}}\rangle =\alpha |0\rangle +\beta {e}^{i(\theta +\varphi )}|1\rangle$$, for any real ϕ. Thus, by considering the result in Eq. (10) and the properties of R(ϕ), one can see that the drain output state becomes $$|{\psi }_{{\rm{out}}}(\varphi )\rangle =R(\varphi )|{\psi }_{{\rm{inp}}}\rangle$$, when we performed the adjustment for $$\omega /(\lambda \sqrt{2\kappa })$$ given by

$$\frac{\omega }{\lambda \sqrt{2\kappa }}=\ell -\frac{\varphi }{\pi } > 0,$$
(11)

where $$\ell$$ must be odd.

For any given ϕ, the expression (11) above shows us how we should make the adjustment in ω, if λ and κ are fixed. As a first important remark of the data bus role in our device, in case we have a physical system in which ω and λ are fixed, for example in cavity QED, we can choose the best fit of the integer κ in order to implement the gate. It is important to note that the equality in (11) can be obtained without any restriction with respect to the weak ($$\lambda N\approx \omega$$) or strong (λN ≈ ω) coupling regime due to the presence of the term $$\ell$$, which can be an odd large or small number.

In conclusion, with the adjustment made in Eq. (11), the final state becomes

$$|{\rm{\Psi }}({t}_{{\rm{e}}{\rm{x}}})\rangle =|{0}_{s},\{{0}_{g}\}\rangle \otimes [\alpha |{0}_{d}\rangle +\beta {e}^{i(\theta +\varphi )}|{1}_{d}\rangle ]=|{0}_{s},\{{0}_{g}\}\rangle \otimes [R(\varphi )|{\psi }_{{\rm{i}}{\rm{n}}{\rm{p}}}\rangle ].$$

Therefore, that the control can be done by simultaneous adjustment of ω, κ and, whenever available, the parameter λ. That shows that the quantum transistor proposed here can be used to make logic quantum gates, in the same way that classical transistors can implement logic classical gates.

### Performance against decoherence

In order to study the performance of our quantum transistor model against the decoherence effects, we will consider that the system is coupled to dissipative reservoirs according to a Lindblad equation32. As shown in Fig. 1, in a quasi realistic scenario, each oscillator of our system is evolving under action of individual thermal baths, where each one is at temperature T and it is constituted by a infinite set of oscillators whose the average value of the frequency is around to ν. In this case, the dynamics of the system can be written15,16,18,20 as

$$\dot{\rho }(t)=\frac{1}{i\hslash }[H,\rho (t)]+{ {\mathcal L} }_{{\rm{se}}}[\rho (t)]+{ {\mathcal L} }_{{\rm{th}}}[\rho (t)],$$
(12)

where the operators are defined by

$$\begin{array}{c}{{\mathscr{L}}}_{{\rm{s}}{\rm{e}}}[\bullet ]=\sum _{k}\frac{{\gamma }_{k}}{2}[2{a}_{k}\bullet {a}_{k}^{\dagger }-\{{a}_{k}^{\dagger }{a}_{k},\bullet \}],\\ \,{{\mathscr{L}}}_{{\rm{t}}{\rm{h}}}[\bullet ]=\sum _{k}\frac{{\gamma }_{k}{\bar{n}}_{k}}{2}[2{a}_{k}\bullet {a}_{k}^{\dagger }-\{{a}_{k}^{\dagger }{a}_{k},\bullet \}+{\rm{h}}.{\rm{c}}.].\end{array}$$
(13)

The operator $${ {\mathcal L} }_{{\rm{se}}}[\bullet ]$$ is associated with spontaneous emission effects while the operator $${ {\mathcal L} }_{{\rm{th}}}[\bullet ]$$ takes into account the dispersion process associated with a thermal reservoir at temperature T ≠ 0. Here γk is the emission rate of the k-th oscillator (k = s, d, 1, 2, 3, …), $${\bar{n}}_{k}$$ is the average number of thermal photons in the k-th reservoir as calculated from the Planck distribution with $$\bar{n}\mathrm{=1/(}{e}^{h\nu /{k}_{{\rm{B}}}T}-\mathrm{1)}$$, and kB is the Boltzmann constant. In particular, it is worth mentioning that non-unitary effects on quantum transistor has been addressed in superconducting quantum transistor models5, where the authors considered the transfer performance of the device against dephasing noise. Thus, the effects of thermal baths on such models is yet a open question.

Since our main interest is in the final state of the drain oscillator, where we will find the quantum information derived from the logic gate result, the state’s fidelity will be computed through $${\mathcal F} =\langle {\psi }_{{\rm{out}}}|\rho ({t}_{{\rm{ex}}})|{\psi }_{{\rm{out}}}\rangle$$, where ρ(tex) is the whole density matrix of the system which comes from the solution of Eq.(12) while |ψout〉 = |0s, {0g}〉 R(ϕ)|ψ〉 is the ideal output state resulting of the logical operation R(ϕ) on a arbitrary input single qubit state $$|\psi \rangle =\alpha |0\rangle +\beta {e}^{i\theta }|1\rangle$$ encoded in drain oscillator.

For the regime of parameters we are considering in this work (ωs = ωd = ω and {λsm} = {λdm} = λ) it is possible to obtain an analytical solution of master equation Eq. (12) for the initial state in which the information is encoded in the source oscillator. If furthermore we also consider that all of data-bus oscillators are at resonance with the source and drain oscillators, that is, κ = N, the fidelity of finding the desired output state from the unit operation R(ϕ), encoded in the drain oscillators, is written as (see Method Section)

$$\begin{array}{ccc}{\mathscr{F}} & = & {[1+\bar{n}(1-{e}^{-\frac{\pi \gamma }{\lambda \sqrt{2\kappa }}})]}^{-(3+\kappa )}[\bar{n}+{\alpha }^{2}+2{\alpha }^{2}(1-{\alpha }^{2}){e}^{-\frac{\pi \gamma }{2\lambda \sqrt{2\kappa }}}\\ & + & \frac{2{(1-{\alpha }^{2})}^{2}}{(1+\bar{n}){e}^{\frac{\pi \gamma }{\lambda \sqrt{2\kappa }}}-\bar{n}}\,+\,{e}^{-\frac{\pi \gamma }{\lambda \sqrt{2\kappa }}}({\alpha }^{2}-1-\bar{n})],\end{array}$$
(14)

where we already used the normalization condition α2 + β2 = 1 and we set ω as provided by Eq. (11). As expected, $${\mathcal F}$$ depends on the initial state and so that it is convenient to define an average value $$\bar{ {\mathcal F} }={\langle {\mathcal F} \rangle }_{\psi }$$ over all initial state |ψ〉. Therefore, we have

$$\bar{{\mathscr{F}}}={[1+\bar{n}(1-{e}^{-\frac{\pi \gamma }{\lambda \sqrt{2\kappa }}})]}^{-(3+\kappa )}\,[\bar{n}+\frac{1}{3}+\frac{4}{15}{e}^{-\frac{\pi \gamma }{2\lambda \sqrt{2\kappa }}}-(\frac{2}{3}+\bar{n}){e}^{-\frac{\pi \gamma }{\lambda \sqrt{2\kappa }}}+\frac{16}{15}\frac{1}{(1+\bar{n}){e}^{\frac{\pi \gamma }{\lambda \sqrt{2\kappa }}}-\bar{n}}].$$
(15)

The non-trivial form of $$\bar{ {\mathcal F} }$$ with respect to the physical parameters involved does not allow us to find optimal strategies to analyze the effects of noisy environment through an analytical approach. For this reason, we consider a numerical study of the behavior of $$\bar{ {\mathcal F} }$$ as given in the density graph shown in Fig. 2. Firstly, in Fig. 2 (top) we present the behavior of $$\bar{ {\mathcal F} }$$ as a function of the dimensionless parameters γ/λ and kBT/, which are associated to the reservoir parameters γ and T, for different values of the data-bus size κ. The range of values considered here for the quantity kBT/ is constrained to temperature range in which $$\bar{n}\le 1$$, which is associated with the validity regime of the solution in Eq. (14). It is important to emphasize the role played by the data-bus in reducing the decoherence effects in our system. It can be seen that for a given range of kBT/, we can use the size of the data-bus as a strategic tool to enhance the performance of the system against the effects of a thermal environment. Second, the Fig. 2 (bottom) presents an analysis of the fidelity $$\bar{ {\mathcal F} }$$, as a function of the dimensionless parameter γ/λ and κ, for different choices to the thermal reservoir temperature through the parameter kBT/. As a complement to previous results, the Fig. 2 (bottom) suggests that we can not increase the data-bus indefinitely in order to get an optimal performance against decoherence. It establishes an optimal relationship between the data-bus size and the thermal reservoir parameters (γ/λ, kBT/), in which we could perform hardware engineering in order to minimize undesired thermal effects on our transistors. Such a result can be seen most clearly through the maximum point on the line separating the regions of density above and below 0.9 in Fig. 2 (bottom) for the parameters kBT/ = 2  10−1, kBT/ = 5  10−1 and kBT/ = 1  100. In addition, from the Eqs (14) and (15), it is noted that the probability of success is independent on the phase-shift gate ϕ that will be implemented in our quantum device. Therefore, the results present in Fig. 2 are valid for any ϕ.

In order to give an experimental notion of how useful can be our transistor against decoherence effect, let us give a realistic example. Firstly, it is important to mention, in the rotating wave and Markov approximations, that the relevant coupling between each oscillator of the transistor with the thermal bath happens when the frequency ν is around to ω33,34, where the characteristic value of ω in several system is of order of about 10 GHz35,36. With this approximate value of ω we can estimate the value of the temperature of the thermal bath from the quantity kBT/ that appears on each graph in Fig. 2 (bottom). In fact, by using the experimental values of the constants37,38 kB = 1.380  10−34 J K−1 and h = 6.626  10−23 J s, for the case in which we have kBT/ = 5  10−1, for example, the temperature obtained will be T ≈ 0.24 K. Thus, by considering the graph in Fig. 2 (bottom), one can conclude that, for the reservoir in which γ/λ ≤ 0.1 and T ≤ 0.24 K, the quantum transistor will work with high fidelity if we design a data-bus with approximately 10 oscillators.

## Discussion

In this paper, we present a quantum transistor model based on quantum oscillators networks. We believe that it can be a useful device for the quantum information processing with optical devices implemented experimentally in both cavity-QED and circuit-QED, for example. Our model explores the frequency detuning between the data-bus oscillators (the gate) with the source and drain oscillators so that the data-bus allow us to create a “potential barrier” to block or transfer the quantum information from the source to the drain. In this sense, the gate oscillators can be seen as an optical quantum switch for quantum information currents. In addition to blocking or transferring quantum information (when the barrier is removed), the transistor proposed here can be used to apply individual quantum gates when the oscillator frequency, the coupling strength between the oscillators and the number of resonant data-bus oscillators is properly adjusted. When considering the inevitable coupling of the system with a thermal reservoir, the performance of the transistor is dictated by the parameters of the environment, namely, the bath temperature T and the emission rate γ. As expected, the system is strongly affected as the temperature T increases. However, we can maintain the high fidelity transfers (as well as the implementation of the phase-shift gate) for cavities with low γ emission rate or high quality factor. In particular, we have shown that the size of the data-bus (κ) can be used as a parameter to control the decoherence effects of the system. In the cases we consider here, we find graphically the existence of an optimal non-trivial criterion for the parameter κ, which depends on both the temperature and the spontaneous emission rate. The knowledge of this criterion allows us to design specific quantum devices where we can enhance the transfer/blocking fidelity against the effects of the thermal bath, in which the temperature and spontaneous emission rate are known. Obviously, the adaptability of our device depends heavily on our experimental ability to turn data-bus quantum oscillators on or off through the atom-field dispersive interaction. Provided that it can be done without too much difficulty, our model can be perfectly adapted to a wide variety of situations imposed by the thermal bath.

Since in our model we are interested in the weak coupling regime between the oscillators (i.e., $$\sqrt{N}\lambda \ll \omega$$), the rotating wave approximation can be performed. In the hypothesis that it is possible to implement the strong coupling regime between the oscillators, we know from literature15,16,18,19,20 that there will be cross-dissipation channels that increase the fidelity of some particular state to be transferred or even eliminate the decoherence effect (dark-states). In view of this result, a natural extension of our work to the strong coupling regime should reveal us some additional gains in fidelity to some initial states and loss to others. As for the entanglement between the source-drain oscillators, what we expect, based on the refs. 15,16,18,19,20, we knows that in a state recurrence time and/or state transfer time the entanglement degree goes to zero, because the states factorize from each other, and it is maximum in half this time, when we have a state entangled with the all data-bus oscillators. The fact that we have a reduction in the short time scale with the increase in the number of resonant data bus oscillators will only tell us that the degree of entanglement reaches its maximum value faster and not that the entanglement increases. This is because the topology of the network, which we propose in this article, is a sum of several transmission lines connected only by the extreme oscillators (each line comprises 3 oscillators). An understanding of the classical point of view can be made here: by increasing the number of these transmission lines, we reduce the fraction of the state to be transmitted between each line, reducing the short time scale and, according to the network adjustment, we can reduce the transfer time and consequently decrease the harmful effects of a thermal reservoir whose time scale remains unchanged.

Throughout this paper, we have studied a device that can be applied to short-range communication, once we are interested in controlling quantum information within quantum devices. However, it is reasonable to believe that our model could be extended to provide long-range communication, where it would require a growth in the number of quantum oscillators or a change in data bus topology. We believe that our proposal opens perspectives for the development of other schemes of optical quantum transistors, or more complex optical devices derived from it. In addition, other approaches to the development of new quantum transistors can be considered from the quantum transistor models mentioned here. For example, the adiabatic quantum transistor model2 uses slow evolutions to accomplish the task of transferring quantum information. In this sense, we can use adiabaticity shortcuts39,40,41 to speed up this task, where we could provide a superadiabatic quantum transistor. Since such STA method can be implemented in an arbitrary finite time42,43, the use of advanced methods of STA44,45,46,47,48,49,50,51,52,53 to develop such quantum devices could be appreciated for superadiabatic quantum computing42,54. In addition, since this extended model could be efficiently implemented using different physical systems21,22,23,24,25,26,27,28, a theoretical and experimental studies will be considered in future researches.

## Methods

For our purposes, we will consider a data bus consisting of a network of N non-interacting oscillators, which, however, each one is coupled with the source and drain oscillator with a real coupling strength {λsm} = {λdm} = λ. In addition, let’s consider that the frequencies of κ data-bus oscillators are in resonance with the frequencies of the source and drain, ωs = ωd = ω, while the others data-bus oscillators have dissonant frequencies $$\tilde{\omega }=\omega +{\rm{\Delta }}$$. Under these conditions, we can obtain analytical expressions that are written in terms of the eigenvalues and eigenvectors of the matrix $${\mathcal H}$$, defined by

$${\mathscr{H}}=(\begin{array}{ccc}\omega & {\boldsymbol{\Lambda }} & 0\\ {{\boldsymbol{\Lambda }}}^{{\rm{\top }}} & {{\mathscr{H}}}^{{\rm{D}}{\rm{B}}} & {{\boldsymbol{\Lambda }}}^{{\rm{\top }}}\\ 0 & {\boldsymbol{\Lambda }} & \omega \end{array}),$$
(16)

where Λ is a 1 × N matrix, whose elements are Λm = λ, ΛT is the transposed matrix of Λ, whereas the square N × N matrix $${ {\mathcal H} }^{{\rm{DB}}}$$ is a diagonal matrix whose elements are defined by $${ {\mathcal H} }_{mn}^{{\rm{DB}}}=\omega {\delta }_{mn}$$, if 1 ≤ m ≤ κ and $${ {\mathcal H} }_{mn}^{{\rm{DB}}}=\tilde{\omega }{\delta }_{mn}$$, if κ < m ≤ N. According to this matrix, the Hamiltonian

$$H=\hslash [\omega (\sum _{\ell =s,d}{a}_{\ell }^{\dagger }{a}_{\ell }+\sum _{m=1}^{\kappa }{a}_{m}^{\dagger }{a}_{m})+\mathop{\omega }\limits^{ \sim }\sum _{m=\kappa +1}^{N}{a}_{m}^{\dagger }{a}_{m}+\lambda \sum _{\ell =s,d}\sum _{m=1}^{N}({a}_{\ell }^{\dagger }{a}_{m}+{a}_{\ell }{a}_{m}^{\dagger })]$$
(17)

can be put into the matrix form as

$$H=(\begin{array}{ccc}{a}_{s}^{\dagger } & \{{a}_{m}^{\dagger }\} & {a}_{d}^{\dagger }\end{array})(\begin{array}{ccc}\omega & {\boldsymbol{\Lambda }} & 0\\ {{\boldsymbol{\Lambda }}}^{{\rm{\top }}} & {{\mathscr{H}}}_{DB} & {{\boldsymbol{\Lambda }}}^{{\rm{\top }}}\\ 0 & {\boldsymbol{\Lambda }} & \omega \end{array})(\begin{array}{c}{a}_{s}\\ \{{a}_{m}\}\\ {a}_{d}\end{array}).$$
(18)

Considering that j and j′ vary from 0 to N + 1, the eigenvalues and orthonormal eigenvectors of $${\mathcal H}$$ can be labeled as follows:

• Regardless of the value of κ we will always have an eigenvalue Ω0 = ω, whose eigenvector ϑ0 has the components $${C}_{j0}=\mathrm{1/}\sqrt{2}$$, if j = 0; $${C}_{j0}=-\,\mathrm{1/}\sqrt{2}$$, if j = N + 1; and Cj0 = 0 for any other value of j.

• When 2 ≤ κ ≤ N we find κ − 1 identical eigenvalues, which can be labeled by j′ as follows: For 1 ≤ j′ ≤ κ − 1, we obtain the eigenvalue $${{\rm{\Omega }}}_{j^{\prime} }=\omega$$, whose eigenvector ϑj has the components $${C}_{jj^{\prime} }=\mathrm{1/}\sqrt{(j+1)j}$$, if 1 ≤ j ≤ j′; $${C}_{jj^{\prime} }=-\,(j-1)/\sqrt{j(j-1)}$$, if j = j′ + 1; and Cjj = 0 for any other value of j.

• When 0 ≤ κ ≤ N − 2 we find N − κ − 1 equal eigenvalues, which will be labeled by j′ as follows: For κ ≤ j′ ≤ N − 2, we have the eigenvalue $${{\rm{\Omega }}}_{j^{\prime} }=\tilde{\omega }$$, whose eigenvector ϑj has the components $${C}_{jj^{\prime} }=\mathrm{1/}\sqrt{(j-\kappa +1)(j-\kappa )}$$, if κ + 1 ≤ j ≤ j′ + 1; $${C}_{jj^{\prime} }=-\,(j-\kappa -1)/\sqrt{(j-\kappa )(j-\kappa -1)}$$, if j = j′ + 2; and Cjj = 0 for any other value of j.

• Setting the parameters Φ = Δ2 + 62, $$\eta ={\rm{\Delta }}[{{\rm{\Delta }}}^{2}+9(N-3\kappa ){\lambda }^{2}]$$, and $$\theta =\frac{1}{3}\arctan (\sqrt{\frac{{{\rm{\Phi }}}^{3}}{{\eta }^{2}}-1})$$, we can write the last three eigenvalues in compact form as

$${{\rm{\Omega }}}_{j^{\prime} }=\omega +\frac{{\rm{\Delta }}}{3}-(1-3{\delta }_{j^{\prime} ,N})\frac{\sqrt{{\rm{\Phi }}}}{3}\,\cos \,\theta +(1-{\delta }_{j^{\prime} ,N}-2{\delta }_{j^{\prime} ,N-1})\sqrt{\frac{{\rm{\Phi }}}{3}}\,\sin \,\theta ,$$
(19)

where $$j^{\prime} =N-1,N,N+1$$. The eigenvector $${\vartheta }_{j^{\prime} }$$ associated with each of these eigenvalues has coefficients defined by: $${C}_{jj^{\prime} }={{\mathscr{N}}}_{j^{\prime} }$$, if j = 0 or j = N + 1; $${C}_{jj^{\prime} }=2\lambda {{\mathscr{N}}}_{j^{\prime} }/({\rm{\Omega }}j^{\prime} -\omega )$$, if 1 ≤ j ≤ κ; and $${C}_{jj^{\prime} }=2\lambda {{\mathscr{N}}}_{j^{\prime} }/({\rm{\Omega }}j^{\prime} -\tilde{\omega })$$, if κ < j ≤ N, where we define the normalization coefficient by

$${{\mathscr{N}}}_{j{\rm{^{\prime} }}}={[2+\kappa {(\frac{2\lambda }{{{\rm{\Omega }}}_{j{\rm{^{\prime} }}}-\omega })}^{2}+(N-\kappa ){(\frac{2\lambda }{{{\rm{\Omega }}}_{j{\rm{^{\prime} }}}-\mathop{\omega }\limits^{ \sim }})}^{2}]}^{-1/2}.$$
(20)

Once the matrix C is obtained, whose columns are the eigenvectors of $${\mathcal H}$$, we can diagonalize the Hamiltonian so that $${{\bf{C}}}^{-1}\cdot {\mathcal H} \cdot {\bf{C}}={ {\mathcal H} }_{D}$$, where the elements of the diagonal matrix $${ {\mathcal H} }_{D}$$ are the eigenvalues defined above. The new A operators, which follow the same canonical commutation rules as the original operators a, are defined by

$${A}_{j^{\prime} }=\sum _{j}\,{C}_{j^{\prime} j}^{-1}{a}_{j},$$
(21)

remembering that C−1 = CT and that we define a0 = as and aN+1 = ad.

In the situation where we have a thermal reservoir coupled to each of the oscillators of our system, according to ref.18, we can write the temporal evolution through the elements of a matrix Θ(t), defined by:

$${{\rm{\Theta }}}_{j^{\prime} j}(t)={e}^{-\gamma t/2}\sum _{l=0}^{N+1}{C}_{j^{\prime} l}{e}^{-i{{\rm{\Omega }}}_{l}t}{C}_{lj}^{-1}.$$
(22)

The diffusion of the system occurs due to the presence of the matrix J(t), which for situations in which the reservoirs are identical, that is, they have the same spontaneous decay rate γj = γ and the same average number of thermal photons $${\bar{n}}_{j}=\bar{n}$$, can be written as

$${J}_{j^{\prime} j}(t)=2\bar{n}(1-{e}^{-\gamma t}){\delta }_{j^{\prime} j}.$$
(23)

If we consider the initial state of the system given by $$|{\psi }_{{\rm{sgd}}}(0)\rangle =({b}_{0}|{0}_{s}\rangle +{b}_{1}|{1}_{s}\rangle )\otimes |\{{0}_{g}\},{0}_{d}\rangle$$, we will verify that the time evolution of the density operator becomes18

$$\begin{array}{rcl}{\rho }_{{\rm{sgd}}}(t) & = & \sum _{m,n=0}^{1}{b}_{m}^{\ast }{b}_{n}\sum _{l=0}^{n}\prod _{\ell =0}^{N+1}\{\sum _{{r}_{\ell },{s}_{\ell },{k}_{\ell }=0}^{\infty }\sum _{{S}_{\ell }=0}^{{\rm{\min }}\,\{{r}_{\ell },{s}_{\ell }\}+{k}_{\ell }}{(-1)}^{{k}_{\ell }}\\ & & \times \,\frac{{[{{\rm{\Theta }}}_{\ell 0}(t)]}^{{r}_{\ell }+{k}_{\ell }-{S}_{\ell }} \% {[{{\rm{\Theta }}}_{\ell 0}^{\ast }(t)]}^{{s}_{\ell }+{k}_{\ell }-{S}_{\ell }}{[\bar{n}(1-{e}^{-\gamma t})]}^{{S}_{\ell }}}{({r}_{\ell }+{k}_{\ell }-{S}_{\ell })!({s}_{\ell }+{k}_{\ell }-{S}_{\ell })!{S}_{\ell }!}\\ & & \times \,\frac{({r}_{\ell }+{k}_{\ell })!({s}_{\ell }+{k}_{\ell })!}{{k}_{\ell }!\sqrt{{r}_{\ell }!{s}_{\ell }!}}|{r}_{\ell }\rangle \langle {s}_{\ell }|\}\delta [m-n+\sum _{\ell =0}^{N+1}({r}_{\ell }-{s}_{\ell })]\delta [l-\sum _{\ell =0}^{N+1}({r}_{\ell }+{k}_{\ell }-{S}_{\ell })],\end{array}$$
(24)

where we are considering the fact that δ[x] = 1 if x = 0 and δ[x] = 0 if x ≠ 0.

## References

1. 1.

Wootters, W. K. & Zurek, W. H. A single quantum cannot be cloned. Nat. 299, 802–803 (1982).

2. 2.

Bacon, D., Flammia, S. T. & Crosswhite, G. M. Adiabatic quantum transistors. Phys. Rev. X 3, 021015 (2013).

3. 3.

Williamson, D. J. & Bartlett, S. D. Symmetry-protected adiabatic quantum transistors. New J. Phys. 17, 053019 (2015).

4. 4.

Marchukov, O. V., Volosniev, A., Valiente, M., Petrosyan, D. & Zinner, N. Quantum spin transistor with a heisenberg spin chain. Nat. Commun. 7, 13070 (2016).

5. 5.

Loft, N. J. S., Kristensen, L. B., Andersen, C. K. & Zinner, N. T. Quantum spin transistors in superconducting circuits. arXiv preprint arXiv:1802.04292 (2018).

6. 6.

Micheli, A., Daley, A. J., Jaksch, D. & Zoller, P. Single atom transistor in a 1d optical lattice. Phys. Rev. Lett. 93, 140408 (2004).

7. 7.

Vaishnav, J. Y., Ruseckas, J., Clark, C. W. & Juzeliūnas, G. Spin field effect transistors with ultracold atoms. Phys. Rev. Lett. 101, 265302 (2008).

8. 8.

Fuechsle, M. et al. A single-atom transistor. Nat. Nanotech. 7, 242 (2012).

9. 9.

Gajdacz, M., Opatrny, T. & Das, K. K. An atomtronics transistor for quantum gates. Phys. Lett. A 378, 1919–1924 (2014).

10. 10.

Chang, D. E., Sørensen, A. S., Demler, E. A. & Lukin, M. D. A single-photon transistor using nanoscale surface plasmons. Nat. Phys. 3, 807 (2007).

11. 11.

Gardelis, S., Smith, C. G., Barnes, C. H. W., Linfield, E. H. & Ritchie, D. A. Spin-valve effects in a semiconductor field-effect transistor: A spintronic device. Phys. Rev. B 60, 7764–7767 (1999).

12. 12.

Chen, W. et al. All-optical switch and transistor gated by one stored photon. Sci. 1237242 (2013).

13. 13.

Hwang, J. et al. A single-molecule optical transistor. Nat. 460, 76 (2009).

14. 14.

Bose, R., Sridharan, D., Kim, H., Solomon, G. S. & Waks, E. Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity. Phys. Rev. Lett. 108, 227402 (2012).

15. 15.

de Ponte, M. A., de Oliveira, M. C. & Moussa, M. H. Y. Decoherence in a system of strongly coupled quantum oscillators. i. symmetric network. Phys. Rev. A 70, 022324 (2004).

16. 16.

de Ponte, M. A., de Oliveira, M. C. & Moussa, M. H. Y. Decoherence in a system of strongly coupled quantum oscillators. ii. central-oscillator network. Phys. Rev. A 70, 022325 (2004).

17. 17.

Chen, B., Peng, Y.-D., Li, Y. & Qian, X.-F. Robust multiple-range coherent quantum state transfer. Sci. Rep. 6, 28886 (2016).

18. 18.

de Ponte, M. A., Mizrahi, S. S. & Moussa, M. H. Y. Networks of dissipative quantum harmonic oscillators: A general treatment. Phys. Rev. A 76, 032101 (2007).

19. 19.

De Ponte, M., Mizrahi, S. S. & Moussa, M. H. Y. Relaxation-and decoherence-free subspaces in networks of weakly and strongly coupled resonators. Annals Phys. 322, 2077–2084 (2007).

20. 20.

De Ponte, M., de Oliveira, M. & Moussa, M. Decoherence in strongly coupled quantum oscillators. Annals Phys. 317, 72–106 (2005).

21. 21.

Hennessy, K. et al. Quantum nature of a strongly coupled single quantum dot–cavity system. Nat. 445, 896 (2007).

22. 22.

Li, G., Allaart, K. & Lenstra, D. Degree of entanglement between excitons in two quantum dots in a cavity. In Visser, T. D., Lenstra, D. & Schouten, H. F. (eds) proc. Symposium IEEE/LEOS Benelux Chapter, 2002, 195–198 (IEEE/LEOS, 2002).

23. 23.

Zhang, Z., Ding, J. & Wang, H.-T. Entanglement and nonlocality in a coupled-cavity system. Photon. Res. 5, 224–232 (2017).

24. 24.

Irish, E. K., Ogden, C. D. & Kim, M. S. Polaritonic characteristics of insulator and superfluid states in a coupled-cavity array. Phys. Rev. A 77, 033801 (2008).

25. 25.

Xiao, Y.-F. et al. Coupled quantum electrodynamics in photonic crystal cavities towards controlled phase gate operations. New J. Phys. 10, 123013 (2008).

26. 26.

Eisert, J., Friesdorf, M. & Gogolin, C. Quantum many-body systems out of equilibrium. Nat. Phys. 11, 124 (2015).

27. 27.

Kaufman, A. M. et al. Quantum thermalization through entanglement in an isolated many-body system. Sci. 353, 794–800 (2016).

28. 28.

Tomita, T., Nakajima, S., Danshita, I., Takasu, Y. & Takahashi, Y. Observation of the mott insulator to superfluid crossover of a driven-dissipative bose-hubbard system. Sci. advances 3, e1701513 (2017).

29. 29.

de Almeida, N., Napolitano, R. & Moussa, M. Phenomenological-operator approach to dissipation in cavity quantum electrodynamics. Phys. Rev. A 62, 033815 (2000).

30. 30.

Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th edn. (Cambridge University Press, New York, NY, USA, 2011).

31. 31.

Barenco, A. et al. Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995).

32. 32.

Lindblad, G. On the generators of quantum dynamical semigroups. Comm. Math. Phys. 48, 119 (1976).

33. 33.

Scully, M. O. & Zubairy, M. S. Quantum optics (1999).

34. 34.

Walls, D. F. & Milburn, G. J. Quantum optics (Springer Science & Business Media, 2007).

35. 35.

Frunzio, L., Wallraff, A., Schuster, D., Majer, J. & Schoelkopf, R. Fabrication and characterization of superconducting circuit qed devices for quantum computation. IEEE transactions on applied superconductivity 15, 860–863 (2005).

36. 36.

Walther, H., Varcoe, B. T., Englert, B.-G. & Becker, T. Cavity quantum electrodynamics. Reports on Prog. Phys. 69, 1325 (2006).

37. 37.

Mohr, P. J., Newell, D. B. & Taylor, B. N. Codata recommended values of the fundamental physical constants: 2014. Rev. Mod. Phys. 88, 035009 (2016).

38. 38.

Newell, D. B. et al. The codata 2017 values of h, e, k, and nA for the revision of the si. Metrol. 55, L13 (2018).

39. 39.

Demirplak, M. & Rice, S. A. Adiabatic population transfer with control fields. J. Phys. Chem. A 107, 9937 (2003).

40. 40.

Demirplak, M. & Rice, S. A. Assisted adiabatic passage revisited. J. Phys. Chem. B 109, 6838 (2005).

41. 41.

Berry, M. Transitionless quantum driving. J. Phys. A: Math. Theor. 42, 365303 (2009).

42. 42.

Santos, A. C. & Sarandy, M. S. Superadiabatic controlled evolutions and universal quantum computation. Sci. Rep. 5, 15775 (2015).

43. 43.

Coulamy, I. B., Santos, A. C., Hen, I. & Sarandy, M. S. Energetic cost of superadiabatic quantum computation. Front. ICT 3, 19 (2016).

44. 44.

Chen, Y.-H. et al. Improving shortcuts to non-hermitian adiabaticity for fast population transfer in open quantum systems. Ann. Phys. (Berlin) 530, 1700247 (2018).

45. 45.

Santos, A. C. & Sarandy, M. S. Generalized shortcuts to adiabaticity and enhanced robustness against decoherence. J. Phys. A: Math. Theor. 51, 025301 (2018).

46. 46.

Hu, C.-K. et al. Experimental implementation of generalized transitionless quantum driving. Opt. Lett. 43, 3136–3139 (2018).

47. 47.

Chen, X., Lizuain, I., Ruschhaupt, A., Guéry-Odelin, D. & Muga, J. G. Shortcut to adiabatic passage in two- and three-level atoms. Phys. Rev. Lett. 105, 123003 (2010).

48. 48.

Chen, Y.-H., Xia, Y., Chen, Q.-Q. & Song, J. Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems. Phys. Rev. A 89, 033856 (2014).

49. 49.

Garejev, N. et al. Third- and fifth-harmonic generation in transparent solids with few-optical-cycle midinfrared pulses. Phys. Rev. A 89, 033846 (2014).

50. 50.

Lu, M., Xia, Y., Shen, L.-T., Song, J. & An, N. B. Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity. Phys. Rev. A 89, 012326 (2014).

51. 51.

Baksic, A., Ribeiro, H. & Clerk, A. A. Speeding up adiabatic quantum state transfer by using dressed states. Phys. Rev. Lett. 116, 230503 (2016).

52. 52.

Huang, B.-H. et al. Fast quantum state engineering via universal su(2) transformation. Phys. Rev. A 96, 022314 (2017).

53. 53.

Chen, Y.-H., Xia, Y., Wu, Q.-C., Huang, B.-H. & Song, J. Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms. Phys. Rev. A 93, 052109 (2016).

54. 54.

Santos, A. C., Silva, R. D. & Sarandy, M. S. Shortcut to adiabatic gate teleportation. Phys. Rev. A 93, 012311 (2016).

## Acknowledgements

We acknowledge financial support from the Brazilian agencies CNPq and the Brazilian National Institute of Science and Technology for Quantum Information (INCT-IQ). Specially, A.C.S. would like to thank Dr. Marcelo Silva Sarandy (IF/UFF) and M.A.P. would like to thank GILY for useful discussions. Both authors would like to express sincere thanks to Gislaine C. Batistela (UNESP), Belita Koiller (IF/UFRJ) and Kita Macario (IF/UFF) for being so solicitous when requested.

## Author information

Both authors Alan C. Santos and M.A. de Ponte conceived the idea, developed the calculations, discussed the results and wrote the manuscript.

Correspondence to M. A. de Ponte or Alan C. Santos.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions


text:
Main Content

Shapley Values for Machine Learning Model

This topic defines Shapley values, describes two available algorithms in the Statistics and Machine Learning Toolbox™ feature that computes Shapley values, provides examples for each, and shows how to reduce the computational cost.

What Is a Shapley Value?

In game theory, the Shapley value of a player is the average marginal contribution of the player in a cooperative game. That is, Shapley values are fair allocations, to individual players, of the total gain generated from a cooperative game. In the context of machine learning prediction, the Shapley value of a feature for a query point explains the contribution of the feature to a prediction (the response for regression or the score of each class for classification) at the specified query point. The Shapley value corresponds to the deviation of the prediction for the query point from the average prediction, due to the feature. For each query point, the sum of the Shapley values for all features corresponds to the total deviation of the prediction from the average.

The Shapley value of the ith feature for the query point x is defined by the value function v:

 ${\phi }_{i}\left({v}_{x}\right)=\frac{1}{M}\sum _{S\subseteq ℳ\\left\{i\right\}}\frac{{v}_{x}\left(S\cup \left\{i\right\}\right)-{v}_{x}\left(S\right)}{\frac{\left(M-1\right)!}{|S|!\left(M-|S|-1\right)!}}$ (1)
• M is the number of all features.

• $ℳ$ is the set of all features.

• |S| is the cardinality of the set S, or the number of elements in the set S.

• vx(S) is the value function of the features in a set S for the query point x. The value of the function indicates the expected contribution of the features in S to the prediction for the query point x.

Shapley Value in Statistics and Machine Learning Toolbox

You can compute Shapley values for a machine learning model by using a `shapley` object. Use the values to interpret the contributions of individual features in the model to the prediction for a query point. You can compute Shapley values in two ways:

• Create a `shapley` object for a machine learning model with a specified query point by using the `shapley` function. The function computes the Shapley values of all features in the model for the query point.

• Create a `shapley` object for a machine learning model by using the `shapley` function, and then compute the Shapley values for a specified query point by using the `fit` function.

Algorithms

`shapley` offers two types of algorithms: interventional, which uses interventional distributions for the value function, and conditional, which uses conditional distributions for the value function. You can specify the algorithm type to use by setting the `Method` name-value argument of the `shapley` function or the `fit` function.

The difference between the two types of algorithms is the definition of the value function. Both types define the value function so that the sum of the Shapley values of a query point over all features corresponds to the total deviation of the prediction for the query point from the average.

`$\sum _{i=1}^{M}{\phi }_{i}\left({v}_{x}\right)=f\left(x\right)-E\left[f\left(x\right)\right].$`

Therefore, the value function vx(S) must correspond to the expected contribution of the features in S to the prediction (f) for the query point x. The algorithms compute the expected contribution by using artificial samples created from the specified data (X). You must provide `X` through the machine learning model input or a separate data input argument when you create a `shapley` object. In the artificial samples, the values for the features in S come from the query point. For the rest of the features (features in Sc, the complement of S), an interventional algorithm generates samples using interventional distributions, whereas a conditional algorithm generates samples using conditional distributions.

Interventional Algorithms

By default, `shapley` uses one of these interventional algorithms: Kernel SHAP [1], Linear SHAP [1], or Tree SHAP [2].

Computing exact Shapley values can be computationally expensive if `shapley` uses all possible subsets S. Therefore, `shapley` estimates the Shapley values by limiting the maximum number of subsets to use for the Kernel SHAP algorithm. For more details, see Computational Cost.

For linear models and tree-based models, `shapley` offers the Linear SHAP and Tree SHAP algorithms, respectively. These algorithms are computationally less expensive and compute exact Shapley values. The algorithms return the same Shapley values that the Kernel SHAP algorithm returns when using all possible subsets. The Linear SHAP and Tree SHAP algorithms are different from the Kernel SHAP algorithm in these ways:

• The Linear SHAP and Tree SHAP algorithms ignore the `ResponseTransform` property (for regression) and the `ScoreTransform` property (for classification) of the machine learning model. That is, the algorithms compute Shapley values based on raw responses or raw scores without applying response transformation or score transformation, respectively. The Kernel SHAP algorithm uses transformed values if the model specifies transformation in the `ResponseTransform` or `ScoreTransform` property.

• The Kernel SHAP algorithm uses observations with missing values for tree models and ensemble models of tree learners. The Linear SHAP and Tree SHAP algorithms cannot handle observations with missing values for any models.

`shapley` selects an algorithm based on the machine learning model type and other specified options:

An interventional algorithm defines the value function of the features in S at the query point x as the expected prediction with respect to the interventional distribution D, which is the joint distribution of the features in Sc:

`${v}_{x}\left(S\right)={E}_{D}\left[f\left({x}_{S},{X}_{{S}^{c}}\right)\right].$`

xS is the query point value for the features in S, and XSc are the features in Sc.

To evaluate the value function vx(S) at the query point x, with the assumption that the features are not highly correlated, `shapley` uses the values in the data X as samples of the interventional distribution D for the features in Sc:

`${v}_{x}\left(S\right)={E}_{D}\left[f\left({x}_{S},{X}_{{S}^{c}}\right)\right]\approx \frac{1}{N}\sum _{j=1}^{N}f\left({x}_{S},{\left({X}_{{S}^{c}}\right)}_{j}\right).$`

N is the number of observations, and (XSc)j contains the values of the features in Sc for the jth observation.

For example, suppose you have three features in X and four observations: (x11,x12,x13), (x21,x22,x23), (x31,x32,x33), and (x41,x42,x43). Assume that S includes the first feature, and Sc includes the rest. In this case, the value function of the first feature evaluated at the query point (x41,x42,x43) is

`${v}_{x}\left(S\right)=\frac{1}{4}\left[f\left({x}_{41},{x}_{12},{x}_{13}\right)+f\left({x}_{41},{x}_{22},{x}_{23}\right)+f\left({x}_{41},{x}_{32},{x}_{33}\right)+f\left({x}_{41},{x}_{42},{x}_{43}\right)\right].$`

An interventional algorithm is computationally less expensive than a conditional algorithm and supports ordered categorical predictors. However, an interventional algorithm requires the feature independence assumption and uses out-of-distribution samples [4]. The artificial samples created with a mix of the query point and the data X can contain unrealistic observations. For example, (x41,x12,x13) might be a sample that does not occur in the full joint distribution of the three features.

Conditional Algorithm

Specify the `Method` name-value argument as `'conditional'` to use the extension to the Kernel SHAP algorithm [3], which is a conditional algorithm.

A conditional algorithm defines the value function of the features in S at the query point x using the conditional distribution of XSc, given that XS has the query point values:

`${v}_{x}\left(S\right)={E}_{{X}_{{S}^{c}}|{X}_{S}={x}_{S}}\left[f\left({x}_{S},{X}_{{S}^{c}}\right)\right].$`

To evaluate the value function vx(S) at the query point x, `shapley` uses nearest neighbors of the query point, which correspond to 10% of the observations in the data X. This approach uses more realistic samples than an interventional algorithm and does not require the feature independence assumption. However, a conditional algorithm is computationally more expensive, does not support ordered categorical predictors, and cannot handle `NaN`s in continuous features. Also, the algorithm might assign a nonzero Shapley value to a dummy feature, which does not contribute to the prediction, if the dummy feature is correlated with an important feature [4].

Specify Computation Algorithm

This example trains a linear classification model and computes Shapley values using an interventional algorithm (`'Method','interventional'`) and then a conditional algorithm (`'Method','conditional'`).

Train Linear Classification Model

Load the `ionosphere` data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad (`'b'`) or good (`'g'`).

`load ionosphere`

Train a linear classification model. For better accuracy of linear coefficients, specify the objective function minimization technique (`'Solver'` name-value argument) as the limited-memory Broyden-Fletcher-Goldfarb-Shanno quasi-Newton algorithm (`'lbfgs'`).

`Mdl = fitclinear(X,Y,'Solver','lbfgs')`
```Mdl = ClassificationLinear ResponseName: 'Y' ClassNames: {'b' 'g'} ScoreTransform: 'none' Beta: [34x1 double] Bias: -3.7100 Lambda: 0.0028 Learner: 'svm' Properties, Methods ```

Compute Shapley Values Using Interventional Algorithm

Compute the Shapley values for the first observation using the Linear SHAP algorithm, which is an interventional algorithm. You do not have to specify the `Method` name-value argument because `'interventional'` is the default.

```queryPoint = X(1,:); explainer1 = shapley(Mdl,X,'QueryPoint',queryPoint); ```

For a classification model, `shapley` computes Shapley values using the predicted class score for each class. Plot the Shapley values for the predicted class by using the `plot` function.

`plot(explainer1)`

The horizontal bar graph shows the Shapley values for the 10 most important variables, sorted by their absolute values. Each value explains the deviation of the score for the query point from the average score of the predicted class, due to the corresponding variable.

For a linear model, `shapley` assumes features are independent from one another and computes the Shapley values from the estimated coefficients (`Mdl.Beta`) [1]. Compute the Shapley values for the positive class (the second class in `Mdl.ClassNames`, `'g'`) directly from the estimated coefficients.

`linearSHAPValues = (Mdl.Beta'.*(queryPoint-mean(X)))';`

Create a table containing the Shapley values computed from `shapley` and the values from the coefficients.

```t = table(explainer1.ShapleyValues.Predictor,explainer1.ShapleyValues.g,linearSHAPValues, ... 'VariableNames',{'Predictor','Values from shapley','Values from coefficients'})```
```t=34×3 table Predictor Values from shapley Values from coefficients _________ ___________________ ________________________ "x1" 0.28789 0.28789 "x2" 0 0 "x3" 0.20822 0.20822 "x4" -0.01998 -0.01998 "x5" 0.20872 0.20872 "x6" -0.076991 -0.076991 "x7" 0.19188 0.19188 "x8" -0.64386 -0.64386 "x9" 0.42348 0.42348 "x10" -0.030049 -0.030049 "x11" -0.23132 -0.23132 "x12" 0.1422 0.1422 "x13" -0.045973 -0.045973 "x14" -0.29022 -0.29022 "x15" 0.21051 0.21051 "x16" 0.13382 0.13382 ⋮ ```

Compute Shapley Values Using Conditional Algorithm

Compute the Shapley values for the first observation using the extension to the Kernel SHAP algorithm, which is a conditional algorithm.

`explainer2 = shapley(Mdl,X,'QueryPoint',queryPoint,'Method','conditional');`

Plot the Shapley values.

`plot(explainer2)`

The two algorithms identify different sets for the 10 most important variables. Only two variables, `x8` and `x22`, are included in both sets.

Computational Cost

The computational cost for Shapley values increases if the number of observations or features is large.

Large Number of Observations

Computing the value function (v) can be computationally expensive if you have a large number of observations, for example, more than 1000. For faster computation, use a smaller sample of the observations when you create a `shapley` object, or run in parallel by specifying `UseParallel` as `true` when you compute the values using the `shapley` or `fit` function. The `UseParallel` option is available when `shapley` uses the Tree SHAP algorithm for an ensemble of trees, the Kernel SHAP algorithm, or the extension to the Kernel SHAP algorithm. Computing in parallel requires Parallel Computing Toolbox™.

Large Number of Features

Computing the summand in Equation 1 for all possible subsets S can be computationally expensive when M (the number of features) is large for the Kernel SHAP algorithm or the extension to the Kernel SHAP algorithm. The total number of subsets to consider is 2M. Instead of computing the summand for all subsets, you can specify the maximum number of subsets by using the `MaxNumSubsets` name-value argument. `shapley` chooses subsets to use based on their weight values. The weight of a subset is proportional to 1/(denominator of the summand), which corresponds to 1 over the binomial coefficient: $1/\left(\begin{array}{c}M-1\\ |S|\end{array}\right)$. Therefore, a subset with a high or low value of cardinality has a large weight value. `shapley` includes the subsets with the highest weight first, and then includes the other subsets in descending order based on their weight values.

Reduce Computational Cost

This example shows how to reduce the computational cost of Shapley values when you have a large number of both observations and features.

Load the sample data set `NYCHousing2015`.

`load NYCHousing2015`

The data set includes 91,446 observations of 10 variables with information on the sales of properties in New York City in 2015. This example uses these variables to analyze the sale prices (`SALEPRICE`).

Preprocess the data set. Convert the `datetime` array (`SALEDATE`) to the month numbers.

`NYCHousing2015.SALEDATE = month(NYCHousing2015.SALEDATE);`

Train a neural network regression model.

`Mdl = fitrnet(NYCHousing2015,'SALEPRICE','Standardize',true);`

Compute the Shapley values of all predictor variables for the first observation. Measure the time required for the computation by using `tic` and `toc`.

```tic explainer1 = shapley(Mdl,'QueryPoint',NYCHousing2015(1,:));```
```Warning: Computation can be slow because the predictor data has over 1000 observations. Use a smaller sample of the training set or specify 'UseParallel' as true for faster computation. ```
`toc`
```Elapsed time is 150.637868 seconds. ```

As the warning message indicates, the computation can be slow because the predictor data has over 1000 observations.

`shapley` provides several options to reduce the computational cost when you have a large number of observations or features:

• Large number of observations — Use a smaller sample of the training data and run in parallel by specifying `UseParallel` as `true`.

• Large number of features — Specify the `MaxNumSubsets` name-value argument to limit the number of subsets included in the computation.

Start a parallel pool.

`parpool;`
```Starting parallel pool (parpool) using the 'Processes' profile ... Connected to parallel pool with 6 workers. ```

Compute the Shapley values again using a smaller sample of the training data and the parallel computing option. Also, specify the maximum number of subsets as `2^5`.

```NumSamples = 5e2; Tbl = datasample(NYCHousing2015,NumSamples,'Replace',false); tic explainer2 = shapley(Mdl,Tbl,'QueryPoint',NYCHousing2015(1,:), ... 'UseParallel',true,'MaxNumSubsets',2^5); toc```
```Elapsed time is 0.844226 seconds. ```

Specifying the additional options reduces the computation time.

References

[1] Lundberg, Scott M., and S. Lee. "A Unified Approach to Interpreting Model Predictions." Advances in Neural Information Processing Systems 30 (2017): 4765–774.

[2] Lundberg, Scott M., G. Erion, H. Chen, et al. "From Local Explanations to Global Understanding with Explainable AI for Trees." Nature Machine Intelligence 2 (January 2020): 56–67.

[3] Aas, Kjersti, Martin Jullum, and Anders Løland. "Explaining Individual Predictions When Features Are Dependent: More Accurate Approximations to Shapley Values." Artificial Intelligence 298 (September 2021).

[4] Kumar, I. Elizabeth, Suresh Venkatasubramanian, Carlos Scheidegger, and Sorelle Friedler. "Problems with Shapley-Value-Based Explanations as Feature Importance Measures." Proceedings of the 37th International Conference on Machine Learning 119 (July 2020): 5491–500.