Protecting quantum gates from control noise
Brif, Constantin
External controls are necessary to enact quantum logic operations, and the inevitable control noise will result in gate errors in a realistic quantum circuit. We investigate the robustness of quantum gates to the random noise in an optimal control field, by utilizing properties of the quantum control landscape that relates the physical objective (in the present case, the quantum gate fidelity) to the applied controls. An approximate result obtained for the statistical expectation value of the gate fidelity in the weak noise regime is shown to be in excellent agreement with direct Monte Carlo sampling over noise process realizations for fidelity values relevant for practical quantum information processing. Using this approximate result, we demonstrate that maximizing the robustness to additive/multiplicative white noise is equivalent to minimizing the total control time/fluence. Also, a genetic optimization algorithm is used to identify controls with improved robustness to a colored noise characterized by its autocorrelation function.
 Okay, so Constantine will give a brief talk about protecting quantum gates from controlled noise. My name is Constantine Brief. I work in Sandia National Labs and this work was done in collaboration with Mr. Grace and Kevin Young in Sandia and with a group of Ershile Robinson Princeton University. Okay. Okay. So let me start with some basic notation. So we consider just a controlled quantum gate which is a basic element of a quantum circuit and small n is the number of qubits in the gate, capital n is the Hilbert space dimension, w is the target unit of transformation that we want to accomplish and u of t is the actual evolution operator at the final time. So we want to produce the same evolution upon any initial state and we want this evolution to be as close as possible to the target. So an external control as we're hearing in many experimental talks yesterday and today, if you want to enact some useful operation onto a quantum system, some control is necessary to do so. So CFT is external control and both Hamiltonian and evolution operators are functionals of the control function. And we measure gate fidelity. It shows us how well the gate was performed. So it's one minus some normalized distance between the actual evolution and the target. And there are different definitions based on how you measure, how you compute this norm. One of two conventional definitions using Hilbert Schmidt norm, you get this or in some cases it will be absolute value instead of real part. But in work that we are doing, I use this definition, however, all else actually will be the same if this one is used because we work very close to the optimal solution, which means very close to fidelity equal to one. So fidelity itself is also the function of control. And let me talk a little bit about the use so-called quantum control landscape. Quantum control landscape is basically the functional dependence of your objective. In this case, it's quantum gate fidelity on the control variables. So it can be continuous control field or some set of control parameters or discretized version in experimental numerical study. So formally, there will be critical point on this landscape which satisfies this condition. And of course, we are looking for an optimum. So for Fliver look actually for the maximum. So we want to find a point on the landscape which are both critical and also have the second derivative, which is a hashen of fidelity with respect to the control field will be negative semi-definite. So if you want some recent review of all this control theory, this is the reference. And actually by doing analysis of this quantum control landscape for this particular problem with given fidelity definition, we can become the following results. So if system is controllable and if we look only so-called regular critical points, there will be one maximum manifold where all optimal controls on this manifold will also this manifold are optimal. They produce unity fidelity. There will be one minimum manifold with zero fidelity. And all other critical points correspond to so-called saddle manifolds. And if we do optimal control search, those can be easily avoided by a good algorithm. So it's quite easy to get on the top of the landscape and find an optimal solution. So in reality there will be an infinite number of optimal solutions. You start with different initial field, you get to different optimal solutions. But in ideal conditions when there is no noise and no coupling to environment, every of those optimal control solution produces unit fidelity. And actually in many cases coupling to control is linear. Mu here is just dipole operator. And in the case when true, the Hessian at the maximum can be evaluated and given by this nice expression. So Mu of t is just the time-dependent dipole operator in the Heisenberg picture. And actually the structure of this Hessian shows us how flat is the top of the control landscape. And the flatness of the top of the control landscape determines how robust is our gate operation to control errors. So now let me introduce noise errors. So in reality when we have control field, it's always will be noisy. And we just assume that we work very close to the optimal solution. So what we want to produce in all fields is C0. But in reality it will be C0 plus some error. And here Z of t will be a random variable. And G of t will be some deterministic function. And I use this notation. So I can account for two important cases. One is additive noise. So G is just one. Noise is just added to the control field. And second important case is multiplicative noise, whereas this G function is just proportioned to the control field itself. So for example in control of semiconductor qubits by voltages, it will be usually additive noise with some additive Johnson noise. In other cases when it's microwave field or laser field which used to control some atom or ion, then noise can be will be multiplicative. So our approach is based on just taking the fidelity as a function of control field and expanding it into the Taylor series assuming that noise is small. So because we work close to the top, it's optimum. So zero order term is just one. And first order term is zero actually for two reasons. One reason is that first derivative on the top is zero. And also if we consider symmetric noise, actually all expectation values of all odd order contributions will be zero as well. So the leading correction will be of second order term. So the second order in the control variable and this is the Hessian compute at the optimal control field. So we consider some random noise process. So the only thing which we know about basically about this control variable is assumed it's symmetric and we assume it has some given autocorrelation function. So because it's random variable in order to estimate the actual fidelity, we need to average over all possible realizations of the noise. So this gives us statistical expectation value of the quantum gate fidelity which I denote by this notation. So when we average over all the noise processes, this just gives us autocorrelation function. So in many cases when you want to simulate system evolution in the presence of noise, you need to do for Monte Carlo simulations which is numerically very, but because we just do this approximate expansion at the weak noise limit, the only thing which contributes from the noise is this autocorrelation function. And this means that we don't need to actually simulate noise through Monte Carlo because autocorrelation function is something which we know. And so from this expression, we see that our goal is to, this additional term it will always be negative because Hessian is negative semi-definite. And we want this absolute value of this term be as small as possible so that fidelity is as close as possible to 1. And in order to do so, in order to protect quantum gate from control noise, we want to minimize this overlap and contribute. So there is an input of control fields which are optimal and different optimal fields will have different Hessians. And also for in the case of multiplicative noise, those g functions will also be dependent on control. So we can search on the top of the landscape through all possible optimal solutions to find those one which minimize the absolute value of this overlap integral and those optimal solution will be more robust than the others. So before I go to this general case, let me consider some important special case which is white noise. So white noise is basically the worst possible situation because, for example, you know, dynamical decoupling, for example, cannot work against noise with zero correlation time. It's like this noise is infinitely fast so you can't really control it. But let's see what happens. So because a correlation function for white noise is this delta function, instead of two integrals, the less we just want integral, however, there is some additional useful property we see that this Hessian is only diagonal part of it. And diagonal elements of the Hessian actually time independent just because trace of the square of any operator is time independent. And so this trace goes out of the integral and we just left with this expression for expected value of fidelity in the presence of white noise in controls. And for additive white noise, this will be all this integral will be just duration of control. It will be just capital T. And for multiplicative white noise, this integral will be just control energy or is control series like to call it fluid. So we can actually compare how good approximation is by looking at against Monte Carlo simulation. So once again, for additive white noise, the error in the gate is just proportion to control time and all these lines are just from this simple approximate formula for different values of noise strengths. And each circle here is obtained by sampling Monte Carlo sampling over 20,000 noise realizations. And you see that our approximation is extremely good basically up to fidelity of 0.9 or gate error of 0.1, which means in the entire area which is of equantment formation processing. And similar thing can be done for multiplicative white noise. It's the same thing just replays the control time by control energy. And once again, agreement between approximate result and Monte Carlo simulation is very good up to in the area of relatively high fidelity. So as I said, when we have additive white noise, gate error is proportion to control time and the only thing which we can do to increase robustness is to minimize control time. And minimizing control time can be done without sacrificing robustness, but only up to the point. There is a critical time below which our gate will be no longer reachable and we did some exploration of this Pareto front for two objectives, two competing objectives. One is maximizing control, maximizing gate fidelity and second is minimizing the control time. So this Pareto front is flat up to critical time, however, this increases very fast. It shows you that it's not very good to operate below the critical time. There were many different studies of time optimal control. We did some additional work here. This preprint was just appeared on archive yesterday. So you can see that the value of critical time depends on what is the target gate. It also depends actually on the global phase of the target gate. And we also studied for two qubit systems how the value of critical time depends on interqubit coupling. And it's 1 over J where J is interqubit coupling for low couplings, whereas this behavior is different for higher coupling strengths. So going back to the case of multiplicative white noise, the gate errors proportional to control energy. So if you want to minimize control energy, the different stories are minimizing control time. So this plot shows you the energy of fluency of the optimal control field versus the same property of initial field from which we start optimization. And you see that if you want to achieve small energy in the optimal field, you need to start with small energy in the initial field. And then optimization search will take you to optimal field with relatively low, as small fluency as needed to actually achieve the target operation. Those different lines obtained for different final times. And you see that generally as time increases, actually the optimal fidelity goes down. However, this change is not monotonic. So this is final time equal 1. And actually this is final time equal 3. It's below and 4 and 6. So this change is not monotonic. And if you look at the dependence of optimal fluids on control time, it oscillates. So it has an envelope which decreases as 1 over T. And this is generic for all target gates. However, those oscillations have amplitude of oscillations in constant for Hadamard gate and Z gate. And amplitude of oscillations decay exponentially for power x and y gate. So let me just say a few words for general case when it's a collared noise, not white noise. We want to minimize this robustness metric. And I would say the only thing that we can also represent this robustness metrics in the frequency domain by using Fourier transform. So S of Z is autocorrelation function, which is Fourier transform, sorry, it's power spectral density of noise, which is Fourier transform of autocorrelation function. And if we see that in the main, if it's white noise, this will be constant. So there is no much you can do about changing this Fourier transform of the Hessian. However, if there is a peak in the power spectrum with a white noise, you want the peak of the Hessian to be as far separated as possible from the peak of the noise. So we want them to be spectrally separated from each other to increase robustness. And finally, I can say that we use some genetic algorithm to minimize this robustness metric k. So this line is obtained when we just use parameterize control field with one frequency and improvement is by factor of two. Or when we make control field more flexible and parameterize it by four.
 Okay, so Constantine will give a brief talk about protecting quantum gates from controlled  noise.  My name is Constantine Brief.  I work in Sandia National Labs and this work was done in collaboration with Mr. Grace and  Kevin Young in Sandia and with a group of Ershile Robinson Princeton University.  Okay.  Okay.  So let me start with some basic notation.  So we consider just a controlled quantum gate which is a basic element of a quantum  circuit and small n is the number of qubits in the gate, capital n is the Hilbert space  dimension, w is the target unit of transformation that we want to accomplish and u of t is the  actual evolution operator at the final time.  So we want to produce the same evolution upon any initial state and we want this evolution  to be as close as possible to the target.  So an external control as we're hearing in many experimental talks yesterday and today,  if you want to enact some useful operation onto a quantum system, some control is necessary  to do so.  So CFT is external control and both Hamiltonian and evolution operators are functionals of  the control function.  And we measure gate fidelity.  It shows us how well the gate was performed.  So it's one minus some normalized distance between the actual evolution and the target.  And there are different definitions based on how you measure, how you compute this norm.  One of two conventional definitions using Hilbert Schmidt norm, you get this or in some cases  it will be absolute value instead of real part.  But in work that we are doing, I use this definition, however, all else actually will  be the same if this one is used because we work very close to the optimal solution, which  means very close to fidelity equal to one.  So fidelity itself is also the function of control.  And let me talk a little bit about the use so-called quantum control landscape.  Quantum control landscape is basically the functional dependence of your objective.  In this case, it's quantum gate fidelity on the control variables.  So it can be continuous control field or some set of control parameters or discretized  version in experimental numerical study.  So formally, there will be critical point on this landscape which satisfies this condition.  And of course, we are looking for an optimum.  So for Fliver look actually for the maximum.  So we want to find a point on the landscape which are both critical and also have the  second derivative, which is a hashen of fidelity with respect to the control field will be  negative semi-definite.  So if you want some recent review of all this control theory, this is the reference.  And actually by doing analysis of this quantum control landscape for this particular problem  with given fidelity definition, we can become the following results.  So if system is controllable and if we look only so-called regular critical points, there  will be one maximum manifold where all optimal controls on this manifold will also this manifold  are optimal.  They produce unity fidelity.  There will be one minimum manifold with zero fidelity.  And all other critical points correspond to so-called saddle manifolds.  And if we do optimal control search, those can be easily avoided by a good algorithm.  So it's quite easy to get on the top of the landscape and find an optimal solution.  So in reality there will be an infinite number of optimal solutions.  You start with different initial field, you get to different optimal solutions.  But in ideal conditions when there is no noise and no coupling to environment, every  of those optimal control solution produces unit fidelity.  And actually in many cases coupling to control is linear.  Mu here is just dipole operator.  And in the case when true, the Hessian at the maximum can be evaluated and given by  this nice expression.  So Mu of t is just the time-dependent dipole operator in the Heisenberg picture.  And actually the structure of this Hessian shows us how flat is the top of the control  landscape.  And the flatness of the top of the control landscape determines how robust is our gate  operation to control errors.  So now let me introduce noise errors.  So in reality when we have control field, it's always will be noisy.  And we just assume that we work very close to the optimal solution.  So what we want to produce in all fields is C0.  But in reality it will be C0 plus some error.  And here Z of t will be a random variable.  And G of t will be some deterministic function.  And I use this notation.  So I can account for two important cases.  One is additive noise.  So G is just one.  Noise is just added to the control field.  And second important case is multiplicative noise, whereas this G function is just proportioned  to the control field itself.  So for example in control of semiconductor qubits by voltages, it will be usually additive  noise with some additive Johnson noise.  In other cases when it's microwave field or laser field which used to control some atom  or ion, then noise can be will be multiplicative.  So our approach is based on just taking the fidelity as a function of control field and  expanding it into the Taylor series assuming that noise is small.  So because we work close to the top, it's optimum.  So zero order term is just one.  And first order term is zero actually for two reasons.  One reason is that first derivative on the top is zero.  And also if we consider symmetric noise, actually all expectation values of all odd order contributions  will be zero as well.  So the leading correction will be of second order term.  So the second order in the control variable and this is the Hessian compute at the optimal  control field.  So we consider some random noise process.  So the only thing which we know about basically about this control variable is assumed it's  symmetric and we assume it has some given autocorrelation function.  So because it's random variable in order to estimate the actual fidelity, we need to average  over all possible realizations of the noise.  So this gives us statistical expectation value of the quantum gate fidelity which I denote  by this notation.  So when we average over all the noise processes, this just gives us autocorrelation function.  So in many cases when you want to simulate system evolution in the presence of noise,  you need to do for Monte Carlo simulations which is numerically very, but because we  just do this approximate expansion at the weak noise limit, the only thing which contributes  from the noise is this autocorrelation function.  And this means that we don't need to actually simulate noise through Monte Carlo because  autocorrelation function is something which we know.  And so from this expression, we see that our goal is to, this additional term it will always  be negative because Hessian is negative semi-definite.  And we want this absolute value of this term be as small as possible so that fidelity is  as close as possible to 1.  And in order to do so, in order to protect quantum gate from control noise, we want to  minimize this overlap and contribute.  So there is an input of control fields which are optimal and different optimal fields will  have different Hessians.  And also for in the case of multiplicative noise, those g functions will also be dependent  on control.  So we can search on the top of the landscape through all possible optimal solutions to find  those one which minimize the absolute value of this overlap integral and those optimal  solution will be more robust than the others.  So before I go to this general case, let me consider some important special case which  is white noise.  So white noise is basically the worst possible situation because, for example, you know,  dynamical decoupling, for example, cannot work against noise with zero correlation  time.  It's like this noise is infinitely fast so you can't really control it.  But let's see what happens.  So because a correlation function for white noise is this delta function, instead of two  integrals, the less we just want integral, however, there is some additional useful property  we see that this Hessian is only diagonal part of it.  And diagonal elements of the Hessian actually time independent just because trace of the  square of any operator is time independent.  And so this trace goes out of the integral and we just left with this expression for  expected value of fidelity in the presence of white noise in controls.  And for additive white noise, this will be all this integral will be just duration of  control.  It will be just capital T. And for multiplicative white noise, this integral will be just control  energy or is control series like to call it fluid.  So we can actually compare how good approximation is by looking at against Monte Carlo simulation.  So once again, for additive white noise, the error in the gate is just proportion to control  time and all these lines are just from this simple approximate formula for different values  of noise strengths.  And each circle here is obtained by sampling Monte Carlo sampling over 20,000 noise realizations.  And you see that our approximation is extremely good basically up to fidelity of 0.9 or gate  error of 0.1, which means in the entire area which is of equantment formation processing.  And similar thing can be done for multiplicative white noise.  It's the same thing just replays the control time by control energy.  And once again, agreement between approximate result and Monte Carlo simulation is very good  up to in the area of relatively high fidelity.  So as I said, when we have additive white noise, gate error is proportion to control  time and the only thing which we can do to increase robustness is to minimize control  time.  And minimizing control time can be done without sacrificing robustness, but only up to the  point.  There is a critical time below which our gate will be no longer reachable and we did some  exploration of this Pareto front for two objectives, two competing objectives.  One is maximizing control, maximizing gate fidelity and second is minimizing the control  time.  So this Pareto front is flat up to critical time, however, this increases very fast.  It shows you that it's not very good to operate below the critical time.  There were many different studies of time optimal control.  We did some additional work here.  This preprint was just appeared on archive yesterday.  So you can see that the value of critical time depends on what is the target gate.  It also depends actually on the global phase of the target gate.  And we also studied for two qubit systems how the value of critical time depends on  interqubit coupling.  And it's 1 over J where J is interqubit coupling for low couplings, whereas this behavior is  different for higher coupling strengths.  So going back to the case of multiplicative white noise, the gate errors proportional  to control energy.  So if you want to minimize control energy, the different stories are minimizing control  time.  So this plot shows you the energy of fluency of the optimal control field versus the same  property of initial field from which we start optimization.  And you see that if you want to achieve small energy in the optimal field, you need to start  with small energy in the initial field.  And then optimization search will take you to optimal field with relatively low, as  small fluency as needed to actually achieve the target operation.  Those different lines obtained for different final times.  And you see that generally as time increases, actually the optimal fidelity goes down.  However, this change is not monotonic.  So this is final time equal 1.  And actually this is final time equal 3.  It's below and 4 and 6.  So this change is not monotonic.  And if you look at the dependence of optimal fluids on control time, it oscillates.  So it has an envelope which decreases as 1 over T. And this is generic for all target  gates.  However, those oscillations have amplitude of oscillations in constant for Hadamard gate  and Z gate.  And amplitude of oscillations decay exponentially for power x and y gate.  So let me just say a few words for general case when it's a collared noise, not white  noise.  We want to minimize this robustness metric.  And I would say the only thing that we can also represent this robustness metrics in  the frequency domain by using Fourier transform.  So S of Z is autocorrelation function, which is Fourier transform, sorry, it's power spectral  density of noise, which is Fourier transform of autocorrelation function.  And if we see that in the main, if it's white noise, this will be constant.  So there is no much you can do about changing this Fourier transform of the Hessian.  However, if there is a peak in the power spectrum with a white noise, you want the peak of the  Hessian to be as far separated as possible from the peak of the noise.  So we want them to be spectrally separated from each other to increase robustness.  And finally, I can say that we use some genetic algorithm to minimize this robustness metric  k.  So this line is obtained when we just use parameterize control field with one frequency  and improvement is by factor of two.  Or when we make control field more flexible and parameterize it by four.
Protecting_quantum_gates_from_control_noise_0_95.png Protecting_quantum_gates_from_control_noise_96_143.png Protecting_quantum_gates_from_control_noise_144_191.png Protecting_quantum_gates_from_control_noise_192_1031.png Protecting_quantum_gates_from_control_noise_1032_1331.png Protecting_quantum_gates_from_control_noise_1332_1583.png Protecting_quantum_gates_from_control_noise_1584_1631.png Protecting_quantum_gates_from_control_noise_1632_1691.png Protecting_quantum_gates_from_control_noise_1692_1739.png Protecting_quantum_gates_from_control_noise_1740_2531.png Protecting_quantum_gates_from_control_noise_2532_4907.png Protecting_quantum_gates_from_control_noise_4908_6671.png Protecting_quantum_gates_from_control_noise_6672_9911.png Protecting_quantum_gates_from_control_noise_9912_17387.png Protecting_quantum_gates_from_control_noise_17388_19427.png Protecting_quantum_gates_from_control_noise_19428_20939.png Protecting_quantum_gates_from_control_noise_20940_21191.png Protecting_quantum_gates_from_control_noise_21192_21371.png Protecting_quantum_gates_from_control_noise_21372_21539.png Protecting_quantum_gates_from_control_noise_21540_23051.png Protecting_quantum_gates_from_control_noise_23052_24311.png Protecting_quantum_gates_from_control_noise_24312_26615.png Protecting_quantum_gates_from_control_noise_26616_27383.png Protecting_quantum_gates_from_control_noise_27384_28715.png Protecting_quantum_gates_from_control_noise_28716_29027.png Protecting_quantum_gates_from_control_noise_29028_29777.png
Okay, so Constantine will give a brief talk about protecting quantum gates from controlled noise. My name is Constantine Brief. I work in Sandia National Labs and this work was done in collaboration with Mr. Grace and Kevin Young in Sandia and with a group of Ershile Robinson Princeton University. Okay. Okay. So let me start with some basic notation. So we consider just a controlled quantum gate which is a basic element of a quantum circuit and small n is the number of qubits in the gate, capital n is the Hilbert space dimension, w is the target unit of transformation that we want to accomplish and u of t is the actual evolution operator at the final time. So we want to produce the same evolution upon any initial state and we want this evolution to be as close as possible to the target. So an external control as we're hearing in many experimental talks yesterday and today, if you want to enact some useful operation onto a quantum system, some control is necessary to do so. So CFT is external control and both Hamiltonian and evolution operators are functionals of the control function. And we measure gate fidelity. It shows us how well the gate was performed. So it's one minus some normalized distance between the actual evolution and the target. And there are different definitions based on how you measure, how you compute this norm. One of two conventional definitions using Hilbert Schmidt norm, you get this or in some cases it will be absolute value instead of real part. But in work that we are doing, I use this definition, however, all else actually will be the same if this one is used because we work very close to the optimal solution, which means very close to fidelity equal to one. So fidelity itself is also the function of control. And let me talk a little bit about the use so-called quantum control landscape. Quantum control landscape is basically the functional dependence of your objective. In this case, it's quantum gate fidelity on the control variables. So it can be continuous control field or some set of control parameters or discretized version in experimental numerical study. So formally, there will be critical point on this landscape which satisfies this condition. And of course, we are looking for an optimum. So for Fliver look actually for the maximum. So we want to find a point on the landscape which are both critical and also have the second derivative, which is a hashen of fidelity with respect to the control field will be negative semi-definite. So if you want some recent review of all this control theory, this is the reference. And actually by doing analysis of this quantum control landscape for this particular problem with given fidelity definition, we can become the following results. So if system is controllable and if we look only so-called regular critical points, there will be one maximum manifold where all optimal controls on this manifold will also this manifold are optimal. They produce unity fidelity. There will be one minimum manifold with zero fidelity. And all other critical points correspond to so-called saddle manifolds. And if we do optimal control search, those can be easily avoided by a good algorithm. So it's quite easy to get on the top of the landscape and find an optimal solution. So in reality there will be an infinite number of optimal solutions. You start with different initial field, you get to different optimal solutions. But in ideal conditions when there is no noise and no coupling to environment, every of those optimal control solution produces unit fidelity. And actually in many cases coupling to control is linear. Mu here is just dipole operator. And in the case when true, the Hessian at the maximum can be evaluated and given by this nice expression. So Mu of t is just the time-dependent dipole operator in the Heisenberg picture. And actually the structure of this Hessian shows us how flat is the top of the control landscape. And the flatness of the top of the control landscape determines how robust is our gate operation to control errors. So now let me introduce noise errors. So in reality when we have control field, it's always will be noisy. And we just assume that we work very close to the optimal solution. So what we want to produce in all fields is C0. But in reality it will be C0 plus some error. And here Z of t will be a random variable. And G of t will be some deterministic function. And I use this notation. So I can account for two important cases. One is additive noise. So G is just one. Noise is just added to the control field. And second important case is multiplicative noise, whereas this G function is just proportioned to the control field itself. So for example in control of semiconductor qubits by voltages, it will be usually additive noise with some additive Johnson noise. In other cases when it's microwave field or laser field which used to control some atom or ion, then noise can be will be multiplicative. So our approach is based on just taking the fidelity as a function of control field and expanding it into the Taylor series assuming that noise is small. So because we work close to the top, it's optimum. So zero order term is just one. And first order term is zero actually for two reasons. One reason is that first derivative on the top is zero. And also if we consider symmetric noise, actually all expectation values of all odd order contributions will be zero as well. So the leading correction will be of second order term. So the second order in the control variable and this is the Hessian compute at the optimal control field. So we consider some random noise process. So the only thing which we know about basically about this control variable is assumed it's symmetric and we assume it has some given autocorrelation function. So because it's random variable in order to estimate the actual fidelity, we need to average over all possible realizations of the noise. So this gives us statistical expectation value of the quantum gate fidelity which I denote by this notation. So when we average over all the noise processes, this just gives us autocorrelation function. So in many cases when you want to simulate system evolution in the presence of noise, you need to do for Monte Carlo simulations which is numerically very, but because we just do this approximate expansion at the weak noise limit, the only thing which contributes from the noise is this autocorrelation function. And this means that we don't need to actually simulate noise through Monte Carlo because autocorrelation function is something which we know. And so from this expression, we see that our goal is to, this additional term it will always be negative because Hessian is negative semi-definite. And we want this absolute value of this term be as small as possible so that fidelity is as close as possible to 1. And in order to do so, in order to protect quantum gate from control noise, we want to minimize this overlap and contribute. So there is an input of control fields which are optimal and different optimal fields will have different Hessians. And also for in the case of multiplicative noise, those g functions will also be dependent on control. So we can search on the top of the landscape through all possible optimal solutions to find those one which minimize the absolute value of this overlap integral and those optimal solution will be more robust than the others. So before I go to this general case, let me consider some important special case which is white noise. So white noise is basically the worst possible situation because, for example, you know, dynamical decoupling, for example, cannot work against noise with zero correlation time. It's like this noise is infinitely fast so you can't really control it. But let's see what happens. So because a correlation function for white noise is this delta function, instead of two integrals, the less we just want integral, however, there is some additional useful property we see that this Hessian is only diagonal part of it. And diagonal elements of the Hessian actually time independent just because trace of the square of any operator is time independent. And so this trace goes out of the integral and we just left with this expression for expected value of fidelity in the presence of white noise in controls. And for additive white noise, this will be all this integral will be just duration of control. It will be just capital T. And for multiplicative white noise, this integral will be just control energy or is control series like to call it fluid. So we can actually compare how good approximation is by looking at against Monte Carlo simulation. So once again, for additive white noise, the error in the gate is just proportion to control time and all these lines are just from this simple approximate formula for different values of noise strengths. And each circle here is obtained by sampling Monte Carlo sampling over 20,000 noise realizations. And you see that our approximation is extremely good basically up to fidelity of 0.9 or gate error of 0.1, which means in the entire area which is of equantment formation processing. And similar thing can be done for multiplicative white noise. It's the same thing just replays the control time by control energy. And once again, agreement between approximate result and Monte Carlo simulation is very good up to in the area of relatively high fidelity. So as I said, when we have additive white noise, gate error is proportion to control time and the only thing which we can do to increase robustness is to minimize control time. And minimizing control time can be done without sacrificing robustness, but only up to the point. There is a critical time below which our gate will be no longer reachable and we did some exploration of this Pareto front for two objectives, two competing objectives. One is maximizing control, maximizing gate fidelity and second is minimizing the control time. So this Pareto front is flat up to critical time, however, this increases very fast. It shows you that it's not very good to operate below the critical time. There were many different studies of time optimal control. We did some additional work here. This preprint was just appeared on archive yesterday. So you can see that the value of critical time depends on what is the target gate. It also depends actually on the global phase of the target gate. And we also studied for two qubit systems how the value of critical time depends on interqubit coupling. And it's 1 over J where J is interqubit coupling for low couplings, whereas this behavior is different for higher coupling strengths. So going back to the case of multiplicative white noise, the gate errors proportional to control energy. So if you want to minimize control energy, the different stories are minimizing control time. So this plot shows you the energy of fluency of the optimal control field versus the same property of initial field from which we start optimization. And you see that if you want to achieve small energy in the optimal field, you need to start with small energy in the initial field. And then optimization search will take you to optimal field with relatively low, as small fluency as needed to actually achieve the target operation. Those different lines obtained for different final times. And you see that generally as time increases, actually the optimal fidelity goes down. However, this change is not monotonic. So this is final time equal 1. And actually this is final time equal 3. It's below and 4 and 6. So this change is not monotonic. And if you look at the dependence of optimal fluids on control time, it oscillates. So it has an envelope which decreases as 1 over T. And this is generic for all target gates. However, those oscillations have amplitude of oscillations in constant for Hadamard gate and Z gate. And amplitude of oscillations decay exponentially for power x and y gate. So let me just say a few words for general case when it's a collared noise, not white noise. We want to minimize this robustness metric. And I would say the only thing that we can also represent this robustness metrics in the frequency domain by using Fourier transform. So S of Z is autocorrelation function, which is Fourier transform, sorry, it's power spectral density of noise, which is Fourier transform of autocorrelation function. And if we see that in the main, if it's white noise, this will be constant. So there is no much you can do about changing this Fourier transform of the Hessian. However, if there is a peak in the power spectrum with a white noise, you want the peak of the Hessian to be as far separated as possible from the peak of the noise. So we want them to be spectrally separated from each other to increase robustness. And finally, I can say that we use some genetic algorithm to minimize this robustness metric k. So this line is obtained when we just use parameterize control field with one frequency and improvement is by factor of two. Or when we make control field more flexible and parameterize it by four.
protecting quantum gates from controlled noise. quantum system, some control is necessary to do and the target. And there are different respect to the control field will be negative quantum control landscape for this particular of the landscape and find an optimal solution. So it's microwave field or laser field which used to consider some random noise process. So the only noise. So this gives us statistical expectation value of the quantum gate fidelity which I denote the weak noise limit, the only thing which autocorrelation function. And this means that we white noise. So white noise is basically the fidelity in the presence of white noise in sampling over 20,000 noise realizations. And you by control energy. And once again, agreement between approximate result and Monte Carlo gate fidelity and second is minimizing the the case of multiplicative white noise, the gate Those different lines obtained for different white noise. We want to minimize this robustness with a white noise, you want the peak of the some genetic algorithm to minimize this