Optimal quantum control with poor statistics

Florian Mintert; Frederic Sauvage

REVIEW 2 major objections 5 minor 102 references

Optimal quantum control with poor statistics

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Bayesian optimization with a binomial noise model finds quantum controls from single-shot data.

desk verdict A useful method paper with a real kernel-choice confound in the key benchmark; the binomial likelihood idea is worth engaging, but the headline advantage is not cleanly shown. read the letter →

arxiv 1909.01229 v3 pith:CIN2XJTY submitted 2019-09-03 quant-ph

Frederic Sauvage , Florian Mintert This is my paper · ORCID

classification quant-ph

keywords quantumoptimalcontrolBayesianoptimizationmeasurementshotnoisesingle-shotmeasurementsGaussianprocesssurrogatebinomiallikelihoodmodel-freeadaptivestrategy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that optimal quantum control does not need accurate estimates of expectation values: if measurement noise is represented by the exact binomial distribution of detector clicks, Bayesian optimization can find good control pulses even when every measurement is a single shot. This matters because model-free control of quantum devices is limited by experimental effort, and repeating measurements to suppress shot noise can erase the benefit of avoiding theory. Across the preparation of a GHZ state, a Mott-insulating phase of a Bose-Hubbard chain, and state preparation on a publicly available quantum processor, the binomial-noise version converges substantially faster than Gaussian-noise Bayesian optimization and faster than a simultaneous-perturbation stochastic approximation baseline. In the GHZ case, binomial modeling reaches infidelities about an order of magnitude lower than Gaussian modeling at equal numbers of runs for $N<1000$, and the single-shot limit still yields percent-level infidelity.

What carries the argument

The load-bearing object is the binomial likelihood of Eq. (5): for a surrogate probability $f(\theta)$, the probability of $n$ successes in $N$ repetitions is $\binom{N}{n} f(\theta)^n (1-f(\theta))^{N-n}$, replacing the phenomenological Gaussian noise model of Eq. (24). It sits inside a Gaussian-process prior with a stationary Matérn kernel (Eqs. (19)-(22)), transformed by the cumulative distribution function of a standard normal so that surrogate landscapes stay in $[0,1]$. The Laplace approximation converts the non-Gaussian posterior over the sampled landscape points into a Gaussian, so the predictive distribution and the acquisition function $a(\theta)=\langle f(\theta)\rangle + \alpha\,\sigma(\theta)$ can be evaluated analytically; this lets the algorithm select the next probe and, through the adaptive strategy of Sec. III B 3, concentrate shots on promising regions.

What would settle it

Run the same binomial Bayesian optimizer on a control landscape that is deliberately discontinuous or has features narrower than the kernel length scale, with single-shot measurements; if its advantage over Gaussian modeling and SPSA disappears or reverses in this regime, the paper's attribution of the speedup to binomial noise modeling would be refuted. A cheaper check is to compare posterior predictive calibration at extreme probabilities near 0 and 1, where the Laplace approximation is known to distort the binomial likelihood.

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Extended reading notes

Core claim

The central claim is that the dominant obstacle to data-driven quantum control is not the sparsity of data but the wrong statistical model of noise. The paper models each measured probability as a Gaussian-process surrogate squeezed into $[0,1]$ by a normal cumulative distribution function, and feeds the actual binomial likelihood of observing $n$ clicks in $N$ repetitions into Bayes' rule; the Laplace approximation keeps the resulting predictive distribution tractable. With this likelihood, an acquisition function balancing expected fidelity and uncertainty selects the next control pulse, and an adaptive schedule starts with few repetitions per measurement and later increases $N$ while shrinking the search domain. The authors report that this procedure finds good controls in the single-shot limit, that it robustly outperforms Gaussian modeling and SPSA in every numerical and experimental example tested, and that the resulting control can then be refined to infidelities near $10^{-8}$ in the GHZ problem.

Load-bearing premise

The algorithm assumes the true control landscape is a smooth sample from a stationary Gaussian process with a Matérn kernel, so that very sparse binary observations can be interpolated reliably; if the real landscape is much rougher or has a different length scale, the acquisition function can be systematically misled before enough data arrives to correct the prior.

Editorial extensions

If this is right

In the GHZ example, fixed-$N$ optimizations show that for any $N$ below 1000, binomial modeling yields infidelities roughly an order of magnitude lower than Gaussian modeling after the same number of runs.
Single-shot measurements ($N=1$) suffice to reach percent-level infidelity, so experimental effort can be shifted from repeating measurements to probing more control pulses.
An adaptive schedule that starts with five repetitions, then raises $N$ and shrinks the parameter domain, refines GHZ infidelities down to about $10^{-8}$ while remaining faster than SPSA at every run budget.
In the presence of unitary noise and readout errors, binomial Bayesian optimization still converges 2 to 5 times faster than SPSA, and a substantial fraction of SPSA runs get trapped in minima with about 50% infidelity.
On the public NISQ device, binomial Bayesian optimization outperformed both Gaussian Bayesian optimization and SPSA in all tested settings, especially with only 300 total runs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: a natural test is to move the binomial likelihood from state-preparation fidelities to gate-learning and variational-energy objectives, where the paper sketches applications but provides no demonstrations; the same counting statistics apply directly.
Editorial extension: the paper's rule that fewer repetitions per point is better suggests a resource-tradeoff curve, since Gaussian-process updates cost $O(M^3)$ in the number of iterations; the paper notes the scaling but does not derive the optimal batch size per pulse.
Editorial extension: in landscapes rougher than the Matérn prior, an adaptive or hierarchical kernel could be needed; comparing fixed versus learned length scales on non-smooth landscapes would separate the contribution of the noise model from that of the smoothness prior.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something genuinely useful: it builds Bayesian optimization for quantum control on an explicit binomial likelihood for measurement shot noise, and shows this can work in the single-shot limit where a Gaussian likelihood essentially fails. The derivations are standard and clear, the numerical benchmarks include medians and quartiles, and the NISQ demonstration is a nice touch. If the central claim were clean, this would be a strong practical contribution to the quantum-control subfield.

But the central claim is not clean. Fig. 4, the key GHZ benchmark that supports the order-of-magnitude advantage, confounds the noise model with the kernel choice: the paper states that binomial modeling uses the Matérn R0 kernel while Gaussian modeling uses R2. So the comparison simultaneously changes the likelihood and the smoothness prior. The advantage attributed to binomial modeling could be partly or entirely due to the rougher R0 prior being a better fit for sparse binary observations. No ablation isolates the likelihood. That is a load-bearing gap, and the stress-test note correctly identifies it. The paper's own Laplace-approximation caveat near 0 and 1 is a secondary but real concern, since the optimal regions are exactly where probabilities saturate.

What the paper does well beyond the benchmark: the single-qubit example uses R2 with binomial modeling and still works, which shows the method is not simply tied to the rougher kernel. The Bose-Hubbard example and the NISQ results are consistent with the claimed trend, though the kernels used there are not stated in the text I read. The comparison against SPSA is also fair and broadly favorable, but SPSA is a weak baseline in many of these landscapes, so that alone does not carry the argument.

The main soft spots are the kernel confound, the hand-picked adaptive schedules with no general principle, and the absence of released code, which makes transfer to new problems harder to assess. The adaptive strategy also switches to Gaussian modeling in later stages, which is a reasonable engineering choice but further blurs the attribution of the overall performance to binomial noise modeling.

Who is this for? Anyone working on closed-loop quantum control with limited measurement budgets will want to know this approach exists. The paper deserves a serious referee, but the referee should demand an ablation or at least a same-kernel comparison before the main claim is accepted. I would not cite the performance claim in its current form, but I would cite the framework once the confound is resolved.

Referee Report

2 major / 5 minor

Summary. The manuscript presents a Bayesian optimization framework for quantum control in which measurement noise is modeled with the exact binomial likelihood rather than with a Gaussian approximation. A warped Gaussian process is used as a prior over each measurement probability, predictions are obtained through a Laplace approximation, and an acquisition function balances the posterior mean and variance. The method is demonstrated on a pedagogical single-qubit landscape, on GHZ-state preparation from a six-parameter circuit, on a Bose-Hubbard filling-error control problem, and on single-qubit state preparation on an IBM Q device. The central claim is that this binomial-noise Bayesian optimization finds high-fidelity controls even with single-shot measurements, and that it substantially outperforms both Gaussian-noise Bayesian optimization and SPSA for the same total number of experimental runs.

Significance. If the central claim holds, this is a practically valuable result: it offers a data-efficient, model-free route to quantum control in the single-shot limit, a regime where standard expectation-value estimation is prohibitively expensive. The paper has several genuine strengths: the statistical formulation is standard and clearly presented; the benchmarks report medians and interquartile ranges over random instances; comparisons are made against a well-known stochastic approximation method; and the NISQ demonstration provides real-device evidence. The main weakness is that the headline GHZ comparison changes two modeling ingredients simultaneously, so the attribution of the improvement to the binomial likelihood is not yet established.

major comments (2)

[III B 2, Fig. 4] The central comparison between binomial and Gaussian modeling is confounded. Section III B 2 states that binomial modeling uses the Matérn kernel R0 (Eq. 20) while Gaussian modeling uses R2 (Eq. 22). The two approaches therefore differ not only in the likelihood but also in the smoothness of the prior over the control landscape. The order-of-magnitude advantage attributed to binomial noise modeling in Fig. 4 could in principle be caused by the rougher R0 prior being better suited to sparse binary observations, or by an interaction between kernel choice and likelihood. Since the paper's abstract and conclusions attribute the improvement to the binomial modeling, an ablation that crosses or matches the kernels (for example, binomial with R2 and Gaussian with R0) is required before that claim is supported.
[III B 1, Eq. (27), Nr = 5MN] The run-count formula is not sufficiently justified. The text says that 'since the three observables S5, S6 and S7 commute,' assessing the fidelity for M control-parameter sets with N repetitions requires Nr = 5MN runs. Because S1, S2, S3, S4, and S5, S6, S7 each have common eigenbases, three measurement settings per control-parameter set would suffice if simultaneous measurements are used, giving Nr = 3MN rather than 5MN. If instead five separate local settings are used (XXX, ZZZ, XYY, YXY, YYX), that should be stated explicitly. The factor directly sets the horizontal axes of Figs. 4, 5, 6, 8 and 9, so the discrepancy matters for every quantitative claim about experimental effort.

minor comments (5)

[III B 2] The text refers to 'the Laplace approximation (Sec. II C),' but the Laplace approximation is introduced in Sec. II F 2 a, not in Sec. II C; the cross-reference should be corrected.
[III A] In the description of the single-qubit example, 'with P2 given in Eq. (22)' should read 'with R2 given in Eq. (22)' to match the polynomial notation introduced in Sec. II F 1.
[III B and III C] The value of the acquisition weight α in Eq. (17) is specified only for the illustrative example of Fig. 2. For the GHZ and Bose-Hubbard benchmarks, α is not reported, which makes the acquisition policy and thus the numerical results hard to reproduce exactly.
[Table I] The bracketed uncertainties in Table I are described as uncertainties of the medians due to the finite number of repetitions, but the method used to compute them (e.g., bootstrap or standard error of the median) is not stated.
[Fig. 4 inset] The inset omits interquartile intervals for visual clarity, but in the single-shot regime a compact indication of the spread, such as min-max whiskers, would help the reader assess the reliability of the reported medians.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed convergence results are validated by independent numerical simulations and a real NISQ device, and the algorithm does not assume its own success.

full rationale

The paper's central claim is that Bayesian optimization with exact binomial modeling of measurement shot noise finds good control solutions even with single-shot measurements. This is established by independent benchmarks: noiseless and noisy numerical simulations of GHZ-state preparation, Bose-Hubbard state preparation, and an IBM quantum-device demonstration (Secs. III B-D, Table I). The algorithm's ingredients are standard Gaussian-process priors (Sec. II F 1), a binomial likelihood derived directly from repeated Bernoulli measurements (Eqs. 5-6), and an acquisition function (Eq. 17); none of these is defined in terms of the target result, so the convergence curves in Figs. 4-10 do not reduce to an input by construction. The single self-citation, Ref. [34], appears only in a background list of Bayesian-optimization applications and is not load-bearing for any derivation. The paper's own caveat about the Laplace approximation (Sec. III B 2) and the hand-chosen adaptive schedules (Sec. III B 3) are limitations on accuracy and generality, not circularity. One methodological confound deserves note without affecting the circularity verdict: in the GHZ benchmark, binomial modeling uses the Matérn kernel R0 while Gaussian modeling uses R2 (Sec. III B 2), so the comparison varies the likelihood and the kernel together; this weakens attribution of the improvement solely to binomial noise modeling, but it is a confound between two independently chosen modeling components, not a self-referential or fitted-input prediction. Overall, the paper is self-contained against external benchmarks and its derivation chain is not circular.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard Bayesian inference and Born-rule statistics. The main assumptions are the adequacy of the GP prior for control landscapes, the independence of surrogate models for multiple observables, and the accuracy of the Laplace approximation. No invented physical entities are introduced.

free parameters (4)

Acquisition weight alpha = 4 in Fig. 2, later 0; other examples unspecified
Balances exploration versus exploitation in the acquisition function Eq. (17); chosen per problem by hand.
GP kernel hyperparameters (variance V, length scale l) = Not reported
Estimated by minimizing the log marginal likelihood (Sec. II F 1); standard, but data-dependent and influential for the surrogate.
Matern kernel polynomial order (R0, R1, R2) = R0 for binomial, R2 for Gaussian in GHZ (Sec. III B 2); other examples not specified
Chosen per example based on the expected roughness of the control landscape.
Adaptive strategy schedule parameters = Initial N=5 or 10; domain reduced around 75 best points; N increased to 100, 250, 500, 50000, 500000
Hand-picked in Sec. III B 3 to demonstrate the strategy; no general selection rule is provided.

assumptions (4)

domain assumption Gaussian process prior with zero mean and Matern kernel encodes the smoothness of the control landscape.
Introduced in Sec. II F 1; if the true landscape is not smooth on the kernel length scale, the surrogate and acquisition function become unreliable.
standard math Measurement outcomes follow independent binomial distributions given the true probabilities.
Follows from Born's rule and independent repetitions, Eqs. (5)-(6).
domain assumption Independent surrogate models for each probability in multi-observable targets.
Eq. (12) treats the f_k independently even when the underlying observables are correlated through the quantum state; possible correlations are ignored.
domain assumption The Laplace approximation to the non-Gaussian posterior is sufficiently accurate.
Sec. II F 2 a; the authors note it degrades near the boundaries 0 and 1 (Sec. III B 2), which is where optima often lie.

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Pith. "Pith review of Optimal quantum control with poor statistics." pith.science (2026). https://pith.science/paper/CIN2XJTY

@misc{pith2026190901229,
  author       = {Pith},
  title        = {Pith review of: Optimal quantum control with poor statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CIN2XJTY}},
  note         = {Machine review of arXiv:1909.01229}
}

read the original abstract

Control of quantum systems is a central element of high-precision experiments and the development of quantum technological applications. Control pulses that are typically temporally or spatially modulated are often designed based on theoretical simulations. As we gain control over larger and more complex quantum systems, however, we reach the limitations of our capabilities of theoretical modeling and simulations, and learning how to control a quantum system based exclusively on experimental data can help us to exceed those limitations. Due to the intrinsic probabilistic nature of quantum mechanics, it is fundamentally necessary to repeat measurements on individual quantum systems many times in order to estimate the expectation value of an observable with good accuracy. Control algorithms requiring accurate data can thus imply an experimental effort that negates the benefits of avoiding theoretical modeling. We present a control algorithm based on Bayesian optimization that finds optimal control solutions in the presence of large measurement shot noise and even in the limit of single-shot measurements. With several numerical and experimental examples we demonstrate that this method is capable of finding excellent control solutions with minimal experimental effort

Figures

Figures reproduced from arXiv: 1909.01229 by the authors.

**Figure 2.** FIG. 2. Three stages of a typical run of Bayesian optimization with extremely bad measurement statistics, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Gate sequence for the preparation of a three [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Convergence of the control algorithms to [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. Comparison of the adaptive strategy (black) [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Convergence to low infidelities in the case [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. Convergence of SPSA in the presence of noise. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. Convergence of the filling error [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: FIG. 9. Optimization results for the adaptive strat [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: FIG. 10. Convergence of optimizations in the presence [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

102 extracted references · 73 canonical work pages

[1]

The prior distribution The prior distribution P (f) describes our lim- ited knowledge on the control landscape before making any observations. In the absence of de- tailed knowledge about a given system, it is chosen based on general, physically reasonable assump- tions: in practice, one would tend to ﬁnd a land- scape that varies smoothly with the contro...
[2]

(2) for our present purposes is the conditional probability density P (D|f)

Expected observations The most relevant term in Eq. (2) for our present purposes is the conditional probability density P (D|f). Even though, one hardly ever knows the true control landscape, one can characterize accu- rately what observations one would expect with a given control landscape. In a perfectly idealized, noiseless situation, one would expect ...
[3]

The prior distribution In order to implement the algorithm laid out so far, one needs to deﬁne the prior distribution P (f) of surrogate models explicitly. Crucially, this does not require an explicit parametrization of possible control landscapes, but this can be done elegantly in terms of Gaussian processes as detailed in the following. A random process...
[4]

(10) requires the solution of anM-dimensional integral

Integration The explicit construction of P (f(θ)|D) in Eq. (10) requires the solution of anM-dimensional integral. Already after a few steps in the iterative control algorithm a numerical evaluation of such an M-dimensional integral is prohibitively expen- sive. An eﬃcient, but still accurate estimate is thus essential for the practical value of the contr...
[5]

Control targets The ﬁdelity F for the state ϱ resulting from the gate sequence is deﬁned as F =⟨Ψ|ϱ|Ψ⟩. Unless one is able to perform a projective measurement in a basis including the state |Ψ⟩, however, one is bound to perform measurements on each individ- ual qubit, as indicated by the three detectors in Fig. 3. In practice, one would therefore construc...
[6]

In all these ex- amples, we will consider convergence in terms of the total number Nr of runs required to reach a given ﬁdelity

Optimization The following optimizations are obtained with R0 (Eq.(20)) for binomial modeling and with R2 (Eq.(22)) for Gaussian modeling. In all these ex- amples, we will consider convergence in terms of the total number Nr of runs required to reach a given ﬁdelity. Since the three observables S5, S6 andS7 commute, an optimization based on the as- sessme...
[7]

Adaptive strategy Given that working with few repetitions of the same measurement (i.e. small values ofN) is ben- eﬁcial for fast convergence, but that this expensive beneﬁt tails oﬀ at high ﬁdelities, one may use a variable number of N for an easily implementable reduction of computational eﬀort. We found it most practical to start the optimization with ...
[8]

To this end, we can con- sider the noisy unitaries Nε = √ 1−ε21 +ε⃗ n⃗ σ, where⃗ σis the vector of the three Pauli matrices

Additional noise Last but not least, we can demonstrate that the present control algorithm is not just able to cope with measurement noise, but that it can also be used to ﬁnd good controls in the presence of addi- tional sources of noise. To this end, we can con- sider the noisy unitaries Nε = √ 1−ε21 +ε⃗ n⃗ σ, where⃗ σis the vector of the three Pauli ma...

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Remarkably, noise can help in the convergence of the algorithm, especially for a low total number of runs (Nr < 10 000)

measurements case. Remarkably, noise can help in the convergence of the algorithm, especially for a low total number of runs (Nr < 10 000). The large interquartile intervals depicted in Fig. 7, however, indicate that more than 25% of the optimizations ended up trapped in local...
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Control function To be amendable to optimization the time- dependent control Γ( t) is parametrized in terms ofS parametersθi corresponding to points equally spaced in time, i.e. Γ(ti) =θi , with ti = i S + 1T . (34) The parameters θi for i = 1,...,S are bound to the interval [...
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Given the rapid growth of the 13 102 103 104 105 Number of runs Nr 10−2 10−1 Filling error N=1 N=5 N=10 N=25 N=50 N=100 Optimal FIG

Control target In principle, one could select a state ﬁdelity as the control target. Given the rapid growth of the 13 102 103 104 105 Number of runs Nr 10−2 10−1 Filling error N=1 N=5 N=10 N=25 N=50 N=100 Optimal FIG. 8. Convergence of the ﬁlling error E of a bosonic chain as ...
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The dashed red line shows the result of the best solution found withEmin≈ 1.6%. Hilbert space with particle number Nb and site number L however, the vast majority of quantum states would yield a negligibly small ﬁdelity, and the state ﬁdelity will fail to identify progress in ...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

The prior distribution The prior distribution P (f) describes our lim- ited knowledge on the control landscape before making any observations. In the absence of de- tailed knowledge about a given system, it is chosen based on general, physically reasonable assump- tions: in practice, one would tend to ﬁnd a land- scape that varies smoothly with the contro...

[2] [2]

(2) for our present purposes is the conditional probability density P (D|f)

Expected observations The most relevant term in Eq. (2) for our present purposes is the conditional probability density P (D|f). Even though, one hardly ever knows the true control landscape, one can characterize accu- rately what observations one would expect with a given control landscape. In a perfectly idealized, noiseless situation, one would expect ...

[3] [3]

The prior distribution In order to implement the algorithm laid out so far, one needs to deﬁne the prior distribution P (f) of surrogate models explicitly. Crucially, this does not require an explicit parametrization of possible control landscapes, but this can be done elegantly in terms of Gaussian processes as detailed in the following. A random process...

[4] [4]

(10) requires the solution of anM-dimensional integral

Integration The explicit construction of P (f(θ)|D) in Eq. (10) requires the solution of anM-dimensional integral. Already after a few steps in the iterative control algorithm a numerical evaluation of such an M-dimensional integral is prohibitively expen- sive. An eﬃcient, but still accurate estimate is thus essential for the practical value of the contr...

[5] [5]

Control targets The ﬁdelity F for the state ϱ resulting from the gate sequence is deﬁned as F =⟨Ψ|ϱ|Ψ⟩. Unless one is able to perform a projective measurement in a basis including the state |Ψ⟩, however, one is bound to perform measurements on each individ- ual qubit, as indicated by the three detectors in Fig. 3. In practice, one would therefore construc...

[6] [6]

In all these ex- amples, we will consider convergence in terms of the total number Nr of runs required to reach a given ﬁdelity

Optimization The following optimizations are obtained with R0 (Eq.(20)) for binomial modeling and with R2 (Eq.(22)) for Gaussian modeling. In all these ex- amples, we will consider convergence in terms of the total number Nr of runs required to reach a given ﬁdelity. Since the three observables S5, S6 andS7 commute, an optimization based on the as- sessme...

[7] [7]

Adaptive strategy Given that working with few repetitions of the same measurement (i.e. small values ofN) is ben- eﬁcial for fast convergence, but that this expensive beneﬁt tails oﬀ at high ﬁdelities, one may use a variable number of N for an easily implementable reduction of computational eﬀort. We found it most practical to start the optimization with ...

[8] [8]

To this end, we can con- sider the noisy unitaries Nε = √ 1−ε21 +ε⃗ n⃗ σ, where⃗ σis the vector of the three Pauli matrices

Additional noise Last but not least, we can demonstrate that the present control algorithm is not just able to cope with measurement noise, but that it can also be used to ﬁnd good controls in the presence of addi- tional sources of noise. To this end, we can con- sider the noisy unitaries Nε = √ 1−ε21 +ε⃗ n⃗ σ, where⃗ σis the vector of the three Pauli ma...