Impact of Temporally Correlated Dephasing Noise on the Fidelity of the 2-Qubit Deutsch-Jozsa Algorithm

Souvik Ghosh

REVIEW 1 major objections 1 minor 22 references

Impact of Temporally Correlated Dephasing Noise on the Fidelity of the 2-Qubit Deutsch-Jozsa Algorithm

T0 review · 1 major / 1 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that 2-qubit Deutsch-Jozsa fidelity depends non-monotonically on noise correlation time, with intermediate memory timescales most damaging, and that Markovian models overestimate fidelity when correlations persist.

desk verdict The paper's central non-monotonicity claim is false: the third noise insertion is a no-op, and the exact fidelity is monotonic in τc. read the letter →

arxiv 2506.05509 v1 pith:GL57SJ2R submitted 2025-06-05 quant-ph cond-mat.mtrl-sci

Souvik Ghosh This is my paper

classification quant-phcond-mat.mtrl-sci MSC 81P68 PACS 03.67.Lx03.65.Yz

keywords dephasingnoisenon-MarkovianOrnstein-UhlenbeckprocessDeutsch-Jozsaalgorithmquantumfidelitytemporalcorrelationsdecoherencesimulation

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

This paper asks whether temporally correlated dephasing noise degrades a basic quantum algorithm in ways that memoryless noise models miss. The author models the environment as an Ornstein-Uhlenbeck process and applies phase errors at three points in a 2-qubit Deutsch-Jozsa circuit with a balanced oracle. The central claim is that the algorithm's fidelity---the probability of measuring the correct answer, $|1\rangle$---is non-monotonic in the noise correlation time $\tau_c$, so an intermediate correlation time can be more damaging than very fast or very slow noise. The paper also claims that a Markovian dephasing channel matched to the same per-step error variance overestimates fidelity whenever $\tau_c$ is not extremely short. If true, this implies that noise characterization for near-term devices must track correlation times, not just average error rates.

What carries the argument

The central object is the Ornstein-Uhlenbeck (OU) process, a stationary Gaussian stochastic process with a finite correlation time $\tau_c$, defined by the update $\nu_{k+1}=\nu_k e^{-\Delta t_{\mathrm{step}}/\tau_c}+\sigma_{\mathrm{OU}}\sqrt{1-e^{-2\Delta t_{\mathrm{step}}/\tau_c}}N_k(0,1)$. Each phase error inserted into the circuit is $\phi_k=\nu_k\Delta t_{\mathrm{step}}$, applied as an $R_z(\phi_k)$ rotation on the query qubit at three fixed circuit locations. This process is the only carrier of temporal correlation in the model: tuning $\tau_c$ at fixed $\sigma_{\mathrm{OU}}$ changes how strongly successive phase errors are aligned, which is what the paper argues produces the non-monotonic fidelity profile. The comparison object is a Markovian phase-damping channel with damping parameter $\lambda_{\mathrm{pd}}=(\Delta t_{\mathrm{step}}\sigma_{\mathrm{OU}})^2$, chosen to match the OU single-step phase variance.

What would settle it

Computing the conditioned success probability of the paper's balanced-oracle circuit as a function of the three phase errors $\phi_1,\phi_2,\phi_3$ gives $\cos^2((\phi_1+\phi_2)/2)$, independent of $\phi_3$; averaging the two correlated Gaussian phases from the OU update yields $P=\frac12+\frac12\exp[-\sigma_{\mathrm{OU}}^2\Delta t_{\mathrm{step}}^2(1+e^{-\Delta t_{\mathrm{step}}/\tau_c})]$, which decreases monotonically with $\tau_c$, and comparing this closed form with Fig. 1 (or deleting the third phase error in the simulation) would settle whether the reported dips are real.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that the success probability of the 2-qubit Deutsch-Jozsa algorithm under Ornstein-Uhlenbeck dephasing has a non-trivial dependence on the noise memory time $\tau_c$. In the simulations, low noise ($\sigma_{\mathrm{OU}}=1.0$) leaves fidelity near 0.99 with only small $\tau_c$ variations, while stronger noise ($\sigma_{\mathrm{OU}}=4.0,5.0$) produces dips at intermediate correlation times, for example a minimum of 0.7871 at $\tau_c=1.0\,\Delta t_{\mathrm{step}}$ for $\sigma_{\mathrm{OU}}=5.0$. The author interprets these dips as a memory effect: when $\tau_c$ is comparable to the roughly $3\,\Delta t_{\mathrm{step}}$ interval over which the three phase insertions act, consecutive phase errors do not average out and can accumulate. The paper further claims that a Markovian phase-damping channel with matched single-step variance predicts higher fidelity for $\tau_c>0.1\,\Delta t_{\mathrm{step}}$, with the gap growing with noise strength, and that even at the shortest simulated $\tau_c$ the OU result is slightly below the Markovian prediction.

Load-bearing premise

The argument assumes that the phase error inserted after the final Hadamard gate contributes to the measured fidelity, so that all three noise insertions count equally; in the paper's own balanced-oracle circuit, that final phase error cannot change the probability of measuring $|1\rangle$, leaving only the first two phase errors to set the outcome.

Editorial extensions

If this is right

For noise strengths $\sigma_{\mathrm{OU}}\ge 2.0$, the simulated fidelity versus $\tau_c$ is non-monotonic, with the deepest minimum at $\sigma_{\mathrm{OU}}=5.0$, $\tau_c=1.0\,\Delta t_{\mathrm{step}}$ (fidelity 0.7871).
A Markovian dephasing model with matched per-step variance is accurate only near $\tau_c=0.1\,\Delta t_{\mathrm{step}}$; for larger $\tau_c$ it overestimates fidelity, and the overestimate grows with noise strength.
Even the shortest simulated correlation time gives slightly lower fidelity than the Markovian channel, indicating residual memory or a distributional mismatch beyond the matched variance.
Benchmarks based only on Markovian error rates, such as $T_2$-type times, may overstate the success of interference-based algorithms in environments with noise correlations.
Noise characterization should include the correlation time or spectral shape of the dephasing environment, not just its strength.

Reading between the lines

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

The paper's circuit admits an exact averaged success-probability formula; deriving it and overlaying it on the numerical curves would show which features of Fig. 1 are robust rather than artifacts of finite sampling.
The same phase-error-at-three-points structure should appear in other single-query interference algorithms, so the correlation-time sensitivity is a general design consideration, not a peculiarity of Deutsch-Jozsa.
An engineered-noise experiment that sweeps $\tau_c$ at fixed $\sigma_{\mathrm{OU}}$ could directly test the predicted worst-case memory time.
Removing the third noise insertion or moving it to the ancilla would isolate which of the three phase points actually controls the fidelity, a direct way to test the paper's 'effective duration of about $3\Delta t_{\mathrm{step}}$' interpretation.

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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

Colleague,

Read this one so you don't have to. The central claim—that the DJ fidelity depends non-monotonically on the OU correlation time τc—is wrong, and the paper's own circuit makes that clear. The third phase insertion after the final Hadamard is a measurement no-op: it multiplies the |1> amplitude by e^{iφ3}, which cancels in the computational-basis probability. The fidelity is exactly 1/2 + 1/2 exp[−σ²Δt²(1+e^{−Δt/τc})], strictly decreasing in τc. So there are no intermediate correlation times that are 'more detrimental.' The reported dips are sampling noise from 100 trajectories, and the paper gives no error bars.

Credit where due: the paper is clearly written, the OU discretization is standard, and it honestly discusses its limitations (single qubit noise, 2-qubit toy). The idea of comparing with an equivalent Markovian channel is sensible. But these virtues don't outweigh the load-bearing flaw.

The specific errors: (1) Section IIC says three noise points, but the third doesn't affect the measurement; Section IV's 'approximately 3Δtstep' is false. (2) The non-monotonicity claim in the abstract, Sec. IIIA, and Sec. V is not supported by the analytic expression or by the data once sampling uncertainty is accounted for. (3) No code or data is provided, so the reported simulations can't be checked. The Markovian matching (Eq. 4) is a crude approximation and isn't derived properly, though that's a minor issue compared to the main one.

The stress-test note's '6σ' deviation is overstated—the correct standard error for 100 independent noise trajectories is about 0.02, not 0.008, so the reported points are consistent with the monotonic curve plus noise. But that doesn't help; it just means the data doesn't support the claim.

This paper doesn't clear the bar for refereeing. The central result is false, and the meaningful content reduces to a textbook calculation. Send it back without referee time. It might be worth using in a journal club as an example of why you check the circuit before trusting simulation curves.

Referee Report

1 major / 1 minor

Summary. This paper numerically studies the fidelity of the 2-qubit Deutsch-Jozsa algorithm (balanced oracle) under dephasing noise modeled as an Ornstein-Uhlenbeck (OU) process, applied as discrete Rz rotations at three specified points in the circuit. The authors vary the noise strength sigma_OU and correlation time tau_c, and report that the fidelity depends non-monotonically on tau_c at higher noise strengths, with certain intermediate correlation times being more detrimental. They also compare with a Markovian phase-damping model matched via Eq. (4) and conclude that the Markovian approximation overestimates fidelity for tau_c > 0.1 Delta t. The results are obtained from Qiskit simulations with Ntraj = 100 and Nshots = 1024, but no code, data, or error bars are provided.

Significance. If the claimed non-monotonic dependence on tau_c were correct, it would be an interesting demonstration that noise memory timescales resonant with algorithmic operations can be particularly harmful, with practical implications for noise characterization and error mitigation. The comparison between correlated and Markovian dephasing is a relevant question. However, the central claim is analytically false for the model described in the paper: the fidelity is an exactly computable, strictly decreasing function of tau_c. The paper therefore does not establish its main finding, and the absence of code, data, and error bars prevents independent verification of the numerical results. The topic is relevant, but the core result is not defensible as presented.

major comments (1)

[Section III and Fig. 1] The numerical evidence for non-monotonicity is not statistically quantified. No error bars, code, or raw data are provided, so the reader cannot assess whether the reported local minima are anything beyond Monte Carlo sampling noise. A rough estimate using the analytic distribution gives a standard error of the mean fidelity of order 0.02 for Ntraj = 100 at the relevant parameters, so the deviations of the reported values from the exact monotonic curve are not statistically significant. More fundamentally, the exact solution in the first comment shows that a genuine non-monotonic dependence is impossible in this model, so the burden is on the authors to identify and fix the implementation error.

minor comments (1)

[Section II.D] Figure 1 shows curves without error bars and without specifying whether the lines are interpolations or guides to the eye. Given that the central claim rests on fine features of these curves, error bars are essential.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fidelity calculation is self-contained; the central non-monotonicity claim conflicts with the paper's own exact model equations, but that is a correctness error, not a derivation-reduction.

full rationale

The paper's derivation chain is self-contained and does not reduce its predictions to its inputs by construction. The OU noise model (Eqs. 1-3) is an openly stated assumption, not an ansatz smuggled in via citation; the references to Uhlenbeck-Ornstein and Gardiner are standard external sources, and there are no self-citations at all. The Markovian comparison parameter in Eq. (4) is derived analytically from the single-step OU variance rather than fitted to the target fidelity, so no fitted input is renamed as a prediction. The claimed non-monotonic dependence of fidelity on tau_c is not definitionally forced; indeed, the exact average fidelity for the described circuit is 1/2 + 1/2 exp[-sigma_OU^2 Delta_t^2 (1 + exp(-Delta_t/tau_c))], which decreases monotonically with tau_c, and the third Rz insertion after the final Hadamard cannot affect the measured P(|1>) for the balanced oracle. These inconsistencies between the reported simulation numbers and the paper's own model point to a serious correctness problem in the central claim, but they are not circularity as defined here: the conclusion is not equivalent to the model's input by construction, nor is it supported by a self-citation chain. The limitations acknowledged in Section IV (noise on only one qubit, discrete noise points, two-qubit restriction) are honest scope statements, not circular moves. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim depends on the modeling choices above. No fitted parameters are used; sigma_OU and tau_c are scanned. The false third-noise assumption is the key failure.

assumptions (3)

domain assumption The Ornstein-Uhlenbeck process is an appropriate model for temporally correlated dephasing noise on a qubit.
The paper assumes Gaussian stationary noise with exponential autocorrelation; this is a standard but nontrivial physical modeling choice (Section II.B).
ad hoc to paper Dephasing noise can be represented as three discrete Rz phase rotations at specific circuit points.
The paper inserts phase errors only at three points rather than continuously during gates (Section II.C).
ad hoc to paper The third phase rotation, applied after the final Hadamard, affects the measured success probability.
This assumption is implicit in the paper's discussion of three effective noise operations (Section IV), but it is false: the probability of measuring |1> is independent of this phase.

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Cite this review

Pith. "Pith review of Impact of Temporally Correlated Dephasing Noise on the Fidelity of the 2-Qubit Deutsch-Jozsa Algorithm." pith.science (2026). https://pith.science/paper/GL57SJ2R

@misc{pith2026250605509,
  author       = {Pith},
  title        = {Pith review of: Impact of Temporally Correlated Dephasing Noise on the Fidelity of the 2-Qubit Deutsch-Jozsa Algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GL57SJ2R}},
  note         = {Machine review of arXiv:2506.05509}
}

abstract

Understanding the influence of realistic noise on quantum algorithms is paramount for the advancement of quantum computation. While often modeled as Markovian, environmental noise in quantum systems frequently exhibits temporal correlations, leading to non-Markovian dynamics that can significantly alter algorithmic performance. This paper investigates the impact of temporally correlated dephasing noise, modeled by the Ornstein-Uhlenbeck (OU) process, on the fidelity of the 2-qubit Deutsch-Jozsa algorithm. We perform numerical simulations using Qiskit, systematically varying the noise strength ($\sigma_{\text{OU}}$) and correlation time ($\tau_c$) of the OU process. Our results demonstrate that the algorithm's fidelity exhibits a non-monotonic dependence on $\tau_c$, particularly at higher noise strengths, with certain intermediate correlation times proving more detrimental than others. We find that a standard Markovian dephasing model, matched to the single-step error variance of the OU process, accurately predicts fidelity only in the limit of very short correlation times. For longer correlation times, the Markovian approximation often overestimates the algorithm's fidelity, failing to capture the complex error dynamics introduced by the noise memory. These findings highlight the necessity of incorporating non-Markovian characteristics for accurate performance assessment of quantum algorithms on near-term devices and underscore the limitations of simpler, memoryless noise models.

Figures

Figures reproduced from arXiv: 2506.05509 by the authors.

**Figure 2.** FIG. 2. Fidelity of the 2-qubit Deutsch-Jozsa algorithm (bal [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Reference graph

Works this paper leans on

22 extracted references · 11 canonical work pages

[1]

After the initial Hadamard gate on the query qubit

work page
[2]

After the oracle Uf operation

work page
[3]

After the final Hadamard gate on the query qubit. At each such point j ∈ {1, 2, 3}, a phase rotation gate Rz(ϕj) = diag(1, eiϕj ) was applied, where ϕj was the j-th value obtained from a single realization of the OU noise sequence {νk} generated as per Eq. (2) and scaled by Eq. (3). Each sequence {ϕj} used for one noisy circuit execution was drawn from a ...

work page
[4]

M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Optimized dynamical decoupling in a model quantum memory, Nature 458, 996 (2009), doi:10.1038/nature07951

doi:10.1038/nature07951 2009
[5]

Breuer and F

H.-P. Breuer and F. Petruccione, *The Theory of Open Quantum Systems* (Oxford University Press, Oxford, 2002)

work page 2002
[6]

Bylander, S

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, Noise spectroscopy through dynamical 6 decoupling with a superconducting flux qubit, Nat. Phys. 7, 565 (2011), doi:10.1038/nphys1994

doi:10.1038/nphys1994 2011
[7]

Cleve, A

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Quan- tum algorithms revisited, Proc. R. Soc. A 454, 339 (1998), doi:10.1098/rspa.1998.0164

arXiv 1998
[8]

de Vega and D

I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017), doi:10.1103/RevModPhys.89.015001

doi:10.1103/revmodphys.89.015001 2017

Show all 22 references

[9]

Deutsch and R

D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation, Proc. R. Soc. A 439, 553 (1992), doi:10.1098/rspa.1992.0167

1992
[10]

C. W. Gardiner, *Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences*, 3rd ed. (Springer, Berlin, Heidelberg, 2004)

2004
[11]

A. B. Klimov, D. Sych, L. L. S´ anchez-Soto, and G. Leuchs, Assessing non-Markovianity, Phys. Rev. A 98, 030102(R) (2018), doi:10.1103/PhysRevA.98.030102

2018 doi
[12]

D. A. Lidar, Lecture notes on the theory of open quantum systems, arXiv:1902.00967 (2019)

2019 arXiv
[13]

J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, and C. Urbina, Decoherence of a superconducting qubit due to bias noise, Phys. Rev. B 67, 094510 (2003), doi:10.1103/PhysRevB.67.094510

2003 doi
[14]

M. A. Nielsen and I. L. Chuang, *Quantum Computa- tion and Quantum Information*, 10th Anniversary ed. (Cambridge University Press, Cambridge, 2010)

2010
[15]

Preskill, Lecture Notes for Ph219/CS219 Quantum Computation (1998-2023), http://www.theory.caltech

J. Preskill, Lecture Notes for Ph219/CS219 Quantum Computation (1998-2023), http://www.theory.caltech. edu/~preskill/ph229/

1998
[16]

Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018), doi:10.22331/q-2018-08- 06-79, arXiv:1801.00862

J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018), doi:10.22331/q-2018-08- 06-79, arXiv:1801.00862

2018 arXiv
[17]

Qiskit contributors, Qiskit: An Open-source Framework for Quantum Computing (2023), https://qiskit.org

2023
[18]

Sza´ nkowski, D

P. Sza´ nkowski, D. S. W¨ atzel, T. Charvat, J. Sperling, J. Peˇ rina Jr., A. Buraczewski, M. Horodecki, K.˙Zyczkowski, and Lukasz Rudnicki, Review of non-Markovianity mea- sures for quantum systems, J. Phys. A: Math. Theor. 50, 063001 (2017), doi:10.1088/1751-8121/aa5455

2017 doi
[19]

B. M. Terhal, Quantum error correction for quan- tum memories, Rev. Mod. Phys. 87, 307 (2015), doi:10.1103/RevModPhys.87.307

2015 doi
[20]

G. E. Uhlenbeck and L. S. Ornstein, On the Theory of the Brownian Motion, Phys. Rev. 36, 823 (1930), doi:10.1103/PhysRev.36.823

1930 doi
[21]

Viola and S

L. Viola and S. Lloyd, Dynamical suppression of deco- herence in two-state quantum systems, Phys. Rev. A 58, 2733 (1998), doi:10.1103/PhysRevA.58.2733

1998 doi
[22]

W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715 (2003), doi:10.1103/RevModPhys.75.715

2003 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

After the initial Hadamard gate on the query qubit

work page

[2] [2]

After the oracle Uf operation

work page

[3] [3]

After the final Hadamard gate on the query qubit. At each such point j ∈ {1, 2, 3}, a phase rotation gate Rz(ϕj) = diag(1, eiϕj ) was applied, where ϕj was the j-th value obtained from a single realization of the OU noise sequence {νk} generated as per Eq. (2) and scaled by Eq. (3). Each sequence {ϕj} used for one noisy circuit execution was drawn from a ...

work page

[4] [4]

M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Optimized dynamical decoupling in a model quantum memory, Nature 458, 996 (2009), doi:10.1038/nature07951

doi:10.1038/nature07951 2009

[5] [5]

Breuer and F

H.-P. Breuer and F. Petruccione, *The Theory of Open Quantum Systems* (Oxford University Press, Oxford, 2002)

work page 2002

[6] [6]

Bylander, S

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, Noise spectroscopy through dynamical 6 decoupling with a superconducting flux qubit, Nat. Phys. 7, 565 (2011), doi:10.1038/nphys1994

doi:10.1038/nphys1994 2011

[7] [7]

Cleve, A

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Quan- tum algorithms revisited, Proc. R. Soc. A 454, 339 (1998), doi:10.1098/rspa.1998.0164

arXiv 1998

[8] [8]

de Vega and D

I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017), doi:10.1103/RevModPhys.89.015001

doi:10.1103/revmodphys.89.015001 2017