Tomography-assisted noisy quantum circuit simulator using matrix product density operators

Heng Fan; Kai Xu; Wei-Guo Ma; Yun-Hao Shi

REVIEW 4 major objections 6 minor 1 cited by

Tomography-assisted noisy quantum circuit simulator using matrix product density operators

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Measured gate noise reproduces a 5-qubit device at 0.999 fidelity

desk verdict A useful, honest integration of real QPT data into an MPDO simulator, with one solid but in-sample five-qubit demonstration; deserves peer review after revisions and data release. read the letter →

arxiv 2508.07610 v1 pith:S6ES32VI submitted 2025-08-11 quant-ph

Wei-guo Ma , Yun-Hao Shi , Kai Xu , Heng Fan This is my paper

classification quant-ph

keywords matrixproductdensityoperatorsquantumprocesstomographynoisycircuitsimulationQAOAcrosstalktensornetworkssuperconductingqubitsMaxCut

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

To simulate a noisy quantum computer faithfully, this paper argues, you should use the device's measured noise rather than an assumed depolarizing-plus-relaxation model. The authors perform quantum process tomography on each two-qubit CZ gate of a five-qubit chain, convert the reconstructed process matrix into Kraus operators, and insert those operators into a matrix product density operator (MPDO) circuit simulator. On a one-depth QAOA circuit run on the Quafu Baiwang device, this QPT-fed simulator matches the experimental probability distribution with Jensen-Shannon divergence 3.64e-4 and state fidelity 0.999. By contrast, a standard noise model whose parameters were fitted to the same physical data gives divergence 2.68e-2 and fidelity 0.945. The point is that instrument-specific, empirically captured noise can be built directly into tensor-network simulation, so noisy-circuit predictions can be tested before running the hardware.

What carries the argument

The machinery is the noise bond in the MPDO representation, together with the QPT-to-Kraus conversion pipeline. In an MPDO, each qubit carries a physical index, left and right entanglement bond indices, and a noise bond linking the qubit to its conjugate copy. A measured gate channel, obtained via QPT's chi matrix, is converted to a Choi matrix, diagonalized into Kraus operators, and reshaped into a rank-3 tensor that contracts directly into the ideal gate tensor. SVD truncation of the entanglement bond chi and the noise bond kappa controls the tensor sizes, with simulation cost roughly proportional to N D $chi^{3}$ $kappa^{3}$.

What would settle it

Take QPT data for the same CZ gates at several times, run the same five-qubit QAOA circuit, and compare the QPT-MPDO prediction with demonstrations performed immediately after each calibration; if the Jensen-Shannon divergence to the demonstration grows as the QPT data age, the reported match is a calibration artifact. A second check: if a CZ gate's reconstructed channel differs when a neighboring gate is active, the independence assumption fails.

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Extended reading notes

Core claim

The central discovery is the collapse of the gap between simulation and experiment when real gate noise is used. For each CZ gate in the circuit, the paper obtains its chi matrix by quantum process tomography, converts it to a Choi matrix, eigendecomposes it into Kraus operators, and attaches those Kraus operators to the gate as a tensor with an extra noise bond. Contracting the circuit as an MPDO yields a probability distribution whose Jensen-Shannon divergence from the Quafu demonstration is 3.64e-4, with state fidelity 0.999. A standard noisy circuit model with depolarizing and thermal relaxation noise, even with parameters optimized to match the device data, gives 2.68e-2 and 0.945. The

Load-bearing premise

The result assumes that the noise on each CZ gate during the actual circuit is the same independent, memoryless, time-invariant channel that QPT measured for that gate beforehand, and that single-qubit gate errors are negligible.

Editorial extensions

If this is right

If the central comparison is right, standard noise models with fitted parameters are insufficient for predicting QAOA output on this class of devices, and device-specific measured channels are needed.
The truncation analysis implies a practical resource rule: small and shallow circuits can be simulated with small bond dimensions, while larger or deeper circuits require proportionally larger chi and kappa.
The method opens a pre-experimental workflow in which current QPT data is used to test whether a proposed variational circuit will actually produce the target state on the real device.
The framework can support error mitigation: with the measured noise channel stored as an MPDO tensor, matrix-product-operator inversion of the noisy process can be applied to remove the dominant errors.
The MaxCut experiments show that real noise suppresses high-probability bitstrings in ways an idealized model misses, so the simulator can quantify how much performance degrades before running hardware.

Reading between the lines

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

A stress test this paper leaves implicit: repeat the same protocol with QPT data taken at several times before one demonstration; if the simulated distribution shifts with QPT age, the reported fidelity is a calibration-time snapshot rather than a stable device property.
The paper characterizes each CZ gate independently, so nearest-neighbor crosstalk during simultaneous gates is only partially captured. A natural extension is to tomograph an entire entangling layer as one process; the comparison would show whether per-gate characterization is the limiting assumption.
Because single-qubit gates were modeled as ideal, an equally natural test is a device where single-qubit error is not negligible. If the two-order-of-magnitude gap closure persists only when two-qubit errors dominate, the method's advantage may be specific to that regime.

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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

What you should know: this is a straightforward, honest engineering paper. The authors take real QPT characterizations of CZ gates on a five-qubit Quafu device, convert the χ matrices to Kraus operators, and feed them into a standard MPDO simulator from Ref. [14]. The headline result is that their QPT-MPDO gives Jensen-Shannon divergence 3.64e-4 against the device output, while a standard depolarizing-plus-relaxation model, even with parameters gradient-fitted to the same data, gives 2.68e-2. That is a genuinely striking gap and a useful data point for anyone doing hardware-aware NISQ simulation.

The paper does several things well. It is transparent about its main simplifications: single-qubit gates are treated as ideal, and the text explicitly warns that QPT data must be current because platforms drift. The truncation study (Fig. 15) is the most solid part, with 1,000 random circuits and error bars showing how χ and κ affect fidelity. The MaxCut simulations are fine as applications.

Soft spots, in proportion. The central validation is one five-qubit circuit, measured once, with no error bars or shot counts reported for the QPT-MPDO versus demonstration comparison. The QPT data and the demonstration are from the same device and essentially the same calibration epoch, so the match is an in-sample consistency check rather than an out-of-sample prediction. The stress-test note is right: this does not establish predictive power for new circuits, later times, or different noise conditions. The standard-model baseline is fitted to the same target distribution, which actually makes the QPT gap more impressive rather than less, but it is still an in-sample comparison. The crosstalk discussion is also weaker than the abstract suggests: the measured QPT is for isolated CZ gates, and the crosstalk effects in Fig. 14 are simulated with an arbitrary RZ(α), not measured. Finally, the conclusion claims the approach handles non-Markovian noise, but QPT characterizations are CPTP Markovian channels; that claim overreaches and should be tempered.

Who this is for: people who simulate noisy circuits and want to see whether real gate tomography is worth the effort. It is an incremental integration, not a conceptual breakthrough. But the demonstration is clean, the limitations are mostly acknowledged, and the result is plausible. It deserves a serious referee, and I would accept it with a request for error bars and shot counts, release of the QPT data and code, one out-of-sample check (e.g., a different circuit or a later calibration), and toning down the non-Markovian and crosstalk claims. I would not cite it in my own work in the next year, but I would not mind seeing it in the literature.

Referee Report

4 major / 6 minor

Summary. The paper presents a classical simulator for noisy quantum circuits based on matrix product density operators (MPDOs), in which gate noise is imported from quantum process tomography (QPT) rather than from idealized depolarizing/thermal-relaxation models. The authors develop the tensor-network conversion from a QPT χ matrix to Kraus operators, apply it to CZ gates on the Quafu Baiwang device, and compare QPT-MPDO simulation with a standard noise model on a five-qubit QAOA state-preparation circuit. They report JSD 3.64e-4 and state fidelity 0.999 for QPT-MPDO versus JSD 2.68e-2 and fidelity 0.945 for the standard model. They also simulate noisy MaxCut QAOA circuits and analyze truncation fidelity for random circuits, concluding that measured gate noise is needed to reproduce device behavior.

Significance. If the results hold, the framework is a potentially useful practical tool for pre-experimental evaluation of NISQ circuits, replacing ad hoc noise models with measured process data. The truncation study in Sec. IV.C (1,000 random circuits) provides useful scaling information for χ and κ, and the pipeline from χ matrix to Kraus operators to MPDO is clearly described. However, the central validation is currently a single in-sample replay, so the significance is contingent on the additional out-of-sample and statistical evidence requested below.

major comments (4)

[Sec. IV.A, Fig. 10] The headline comparison rests on one five-qubit QAOA circuit. No shot counts, error bars, or repeated runs are reported, and the QPT data were taken on the same device immediately before the demonstration ('Each independent qubit-interval CZ gate is individually characterized by QPT before operation'). A close match under these conditions is a consistency check, not a predictive test. To support the claim that measured gate noise is needed, validate on circuits not used in the calibration and on data taken at different times or after drift; Sec. V itself notes 'QPT data should be as current as possible.'
[Sec. IV.A, paragraph after Fig. 10] The standard-noise baseline is fit to the same demonstration data by gradient descent using JSD as the loss. The comparison therefore conflates model capacity with physical accuracy: a flexible but inaccurate model can appear better after fitting to the target. Report the optimized parameters (depolarizing rates, thermal relaxation rates, SPAM probability) and either use independently calibrated parameters or perform a cross-validation split; otherwise the 70-fold JSD gap is not interpretable.
[Sec. III and Sec. IV.A] The QPT matrices for the four CZ gates are not included in the paper or a supplement, and the exact χ/κ truncation values for the five-qubit run are not stated. Since the whole method rests on these experimentally measured channels, omitting them makes the central result unreproducible. The authors should deposit the χ matrices (or Kraus operators) and the truncation parameters/errors.
[Sec. V] The conclusion states that the approach 'addresses both Markovian and non-Markovian noise,' but the implementation models each CZ gate as a single memoryless CPTP channel from QPT (Eqs. 12–16). No non-Markovian effect is characterized or simulated in the demonstrations. This claim should be removed or substantiated, e.g., by QPT at multiple time scales.

minor comments (6)

[Table I] The 'CZ gate error rate' entries (0.957, 0.970, 0.980, 0.950) are likely gate fidelities, not error rates; an error rate above 0.95 would make the observed agreement impossible. Please relabel or correct.
[Fig. 10] The text refers to 'pink bars' for demonstration data while the caption says 'orange bars'; also the color names for standard and QPT-MPDO simulations differ between text and caption. Please align the wording and the figure colors.
[Sec. II.C] The notation T[b] and T[p] for the two conjugate tensors is introduced but not defined clearly; the superscripts [b] and [p] are used before their meaning is stated. Consider clarifying with an explicit example or by using +/− labels.
[Eq. (17) and Fig. 8] The text describes a target qubit plus 2N measurement qubits, but the five-qubit circuit in Fig. 8 is not explicitly mapped to N. Specify the relation, e.g., N=2 for the demonstration.
[Sec. IV.A, sentence after Fig. 10] The sentence 'The improved numerical simulation accuracy, however, still surpassed the performance of the real device' seems to say the opposite of what is meant; likely 'failed to match the performance of the real device.' Please correct.
[Sec. IV.C, Fig. 14(b)] The crosstalk angle α is 'arbitrarily chosen' rather than taken from QPT or device calibration. This is fine as a synthetic stress test, but the text should state clearly that crosstalk is not experimentally characterized in this work.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: QPT-MPDO validation rests on independently measured gate processes, not on fitting the target output.

full rationale

The paper's claimed derivation chain is: (1) independently characterize each CZ gate with QPT, converting the process matrix to Kraus operators (Eqs. 12-16); (2) insert those Kraus operators as noisy two-qubit gates in an MPDO circuit simulator (Eqs. 5-8 and Fig. 6); (3) simulate the five-qubit QAOA circuit and compare to the Quafu demonstration. The QPT data come from individual gate characterizations ('Each independent qubit-interval CZ gate is individually characterized by QPT before operation'), not from fitting the QAOA output distribution. In contrast, the paper explicitly states that the standard-noise baseline was fitted to the demonstration data: 'we further used gradient descent ... to optimize the parameters to find a set of ‘optimal’ parameters that are closest to the demonstration data.' Thus the QPT-MPDO result is not a fitted-input-called-prediction; the better agreement of QPT-MPDO relative to the fitted standard model is a genuine comparison between a gate-level independent input and a circuit-level output. The main weakness—that QPT and the demonstration come from the same device and calibration epoch—is acknowledged by the authors themselves ('experimental platforms can change dynamically' and 'the QPT data should be as current as possible'), and it is a generalization limitation, not a circularity: the simulated distribution is not equal to the measured one by construction, since the QPT data alone do not determine the QAOA outcome without the intervening circuit simulation. There is no load-bearing self-citation, no imported uniqueness theorem, and no ansatz smuggled in via citation. The central claim retains independent content, so the appropriate circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its free parameters are the tensor-network truncation dimensions chi and kappa, plus the fitted standard-noise baseline and an arbitrary crosstalk angle. The main assumptions are that QPT gives a complete and stable picture of each gate's noise, that single-qubit gates are ideal, and that the MPDO framework from Ref. [14] works as advertised.

free parameters (4)

MPDO bond dimension chi = unreported in main text
Controls truncation of quantum entanglement bonds. Fig. 15 scans chi and kappa, showing fidelity depends on it. The values used for Fig. 10 are not stated.
MPDO noise bond dimension kappa = unreported in main text
Controls the number of kept Kraus or noise singular values. The central QPT-MPDO result in Fig. 10 depends on a truncation that is not quantified.
Standard-noise model parameters (depolarizing rates, thermal relaxation rates, SPAM probability) = optimized by gradient descent with JSD loss
Section IV.A fits these parameters to the demonstration data to build the baseline. The baseline is therefore a fit, not a predictive model.
Crosstalk RZ angle alpha = random in (10^-5 pi, 10^-3 pi]
Section IV.C arbitrarily chooses this angle to emulate crosstalk. It is not measured for the device and is only used in the separate truncation study.

assumptions (5)

domain assumption QPT characterization of each CZ gate yields a CPTP channel that fully captures the gate's noise in the circuit context.
Section III relies on this to convert measured chi matrices into Kraus operators via Eqs. (12) to (16). If QPT misses non-Markovian or context-dependent effects, the simulator's output could match by coincidence.
domain assumption Single-qubit gate errors are negligible and can be modeled as ideal.
Section III states: 'we performed the single-qubit gates as ideal gates.' The claimed 0.999 fidelity depends on this being a good approximation for the Quafu device.
domain assumption Gate behavior during the QAOA experiment is identical to behavior during the separate QPT characterization.
Section V warns that 'QPT data should be as current as possible.' Device drift or crosstalk during simultaneous operation could break this assumption.
standard math Truncating local tensors by SVD gives a faithful global approximation of the circuit density matrix.
Eq. (10) gives a local truncation error bound, but the paper does not prove a global error bound for the full circuit after repeated contractions and truncations.
standard math The MPDO representation, canonicalization, and noise-bond construction from Ref. [14] are correct and applicable.
The paper imports this machinery without re-deriving it, which is reasonable but means the central simulation framework rests on prior work.

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Cite this review

Pith. "Pith review of Tomography-assisted noisy quantum circuit simulator using matrix product density operators." pith.science (2026). https://pith.science/paper/S6ES32VI

@misc{pith2026250807610,
  author       = {Pith},
  title        = {Pith review of: Tomography-assisted noisy quantum circuit simulator using matrix product density operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6ES32VI}},
  note         = {Machine review of arXiv:2508.07610}
}

read the original abstract

In recent years, efficient quantum circuit simulations incorporating ideal noise assumptions have relied on tensor network simulators, particularly leveraging the matrix product density operator (MPDO) framework. However, experiments on real noisy intermediate-scale quantum (NISQ) devices often involve complex noise profiles, encompassing uncontrollable elements and instrument-specific effects such as crosstalk. To address these challenges, we employ quantum process tomography (QPT) techniques to directly capture the operational characteristics of the experimental setup and integrate them into numerical simulations using MPDOs. Our QPT-assisted MPDO simulator is then applied to explore a variational approach for generating noisy entangled states, comparing the results with standard noise numerical simulations and demonstrations conducted on the Quafu cloud quantum computation platform. Additionally, we investigate noisy MaxCut problems, as well as the effects of crosstalk and noise truncation. Our results provide valuable insights into the impact of noise on NISQ devices and lay the foundation for enhanced design and assessment of quantum algorithms in complex noise environments.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

Works this paper leans on

47 extracted references · 29 canonical work pages · cited by 1 Pith paper

[14]

K. Noh, L. Jiang, and B. Fefferman, Efficient classical simulation of noisy random quantum circuits in one di- mension, Quantum 4, 318 (2020)

work page 2020
[1]

Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018)

J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018)

2018
[2]

Cheng, X.-H

B. Cheng, X.-H. Deng, X. Gu, Y. He, G. Hu, P. Huang, J. Li, B.-C. Lin, D. Lu, Y. Lu et al., Noisy intermediate- scale quantum computers, Frontiers of Physics 18, 21308 (2023)

work page 2023
[3]

#"#!!! !!! $

(4) B. Noise in Circuit Model Quantum circuits are inherently susceptible to various forms of noise that can significantly affect their perfor- mance. It is crucial to set up models that simplify the representation of quantum noise and help manage this complexity. One such model, the unified noisy circuit model [31], begins with two qubits in the state |0...

work page
[4]

Aaronson, Quantum computing, postselection, and probabilistic polynomial-time, Proc

S. Aaronson, Quantum computing, postselection, and probabilistic polynomial-time, Proc. R. Soc. A 461, 3473 (2005)

work page 2005
[5]

M. J. Bremner, R. Jozsa, and D. J. Shepherd, Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy, Proc. R. Soc. A467, 459 (2011)

work page 2011
[6]

M. J. Bremner, A. Montanaro, and D. J. Shepherd, Average-case complexity versus approximate simulation of commuting quantum computations, Phys. Rev. Lett. 117, 080501 (2016)

2016
[7]

Y. Zhou, E. M. Stoudenmire, and X. Waintal, What lim- its the simulation of quantum computers?, Phys. Rev. X 10, 041038 (2020)

2020

Show all 47 references

[8]

Vidal, Efficient simulation of one-dimensional quan- tum many-body systems, Phys

G. Vidal, Efficient simulation of one-dimensional quan- tum many-body systems, Phys. Rev. Lett. 93, 040502 (2004)

2004
[9]

Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue

U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue

2011
[10]

Fannes, B

M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Communica- tions in Mathematical Physics 144, 443 (1992)

1992
[11]

Okunishi, T

K. Okunishi, T. Nishino, and H. Ueda, Developments in the tensor network — from statistical mechanics to quantum entanglement, Journal of the Physical Society of Japan 91, 062001 (2022)

2022
[12]

Pirvu, V

B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, Matrix product operator representations, New Journal of Physics 12, 025012 (2010)

2010
[13]

Verstraete, J

F. Verstraete, J. J. Garc ´ ıa-Ripoll, and J. I. Cirac, Ma- trix product density operators: Simulation of finite- temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004)

2004
[15]

Cheng, C

S. Cheng, C. Cao, C. Zhang, Y. Liu, S.-Y. Hou, P. Xu, and B. Zeng, Simulating noisy quantum circuits with ma- trix product density operators, Physical Review Research 3, 023005 (2021)

2021
[16]

T. B. Wahl and S. Strelchuk, Simulating quantum cir- cuits using efficient tensor network contraction algo- rithms with subexponential upper bound, Phys. Rev. Lett. 131, 180601 (2023)

2023
[17]

I. L. Markov and Y. Shi, Simulating quantum computa- tion by contracting tensor networks, SIAM Journal on Computing 38, 963 (2008)

2008
[18]

B. M. Terhal and D. P. DiVincenzo, Adaptive quan- tum computation, constant depth quantum circuits and Arthur-Merlin games, Quantum Information & Compu- tation 4, 134–145 (2004)

2004
[19]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, 2010)

2010
[20]

Torlai, C

G. Torlai, C. J. Wood, A. Acharya, G. Carleo, J. Car- rasquilla, and L. Aolita, Quantum process tomography with unsupervised learning and tensor networks, Nature Communications 14, 2858 (2023)

2023
[21]

Bharti, A

K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke et al. , Noisy intermediate- scale quantum algorithms, Reviews of Modern Physics 94, 015004 (2022)

2022
[22]

Huang, X.-Y

H.-L. Huang, X.-Y. Xu, C. Guo, G. Tian, S.-J. Wei, X. Sun, W.-S. Bao, and G.-L. Long, Near-term quantum computing techniques: variational quantum algorithms, error mitigation, circuit compilation, benchmarking and classical simulation, Science China Physics, Mechanics & Astron...

2023
[23]

S. Lee, J. Lee, H. Zhai, Y. Tong, A. M. Dalzell, A. Ku- mar, P. Helms, J. Gray, Z.-H. Cui, W. Liu et al. , Eval- uating the evidence for exponential quantum advantage in ground-state quantum chemistry, Nature Communi- cations 14, 1952 (2023)

1952
[24]

McArdle, S

S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, 14 and X. Yuan, Quantum computational chemistry, Re- views of Modern Physics 92, 015003 (2020)

2020
[25]

Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Hug- gins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Reviews of Modern Physics 95, 045005 (2023)

2023
[26]

Jaschke, S

D. Jaschke, S. Montangero, and L. D. Carr, One- dimensional many-body entangled open quantum sys- tems with tensor network methods, Quantum Science and Technology 4, 013001 (2018)

2018
[27]

Zwolak and G

M. Zwolak and G. Vidal, Mixed-state dynamics in one- dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm, Phys. Rev. Lett. 93, 207205 (2004)

2004
[28]

J. Cui, J. I. Cirac, and M. C. Ba˜ nuls, Variational ma- trix product operators for the steady state of dissipative quantum systems, Phys. Rev. Lett. 114, 220601 (2015)

2015
[29]

Kraus, A

K. Kraus, A. B¨ ohm, J. D. Dollard, and W. H. Woot- ters, eds., States, Effects, and Operations: Fundamen- tal Notions of Quantum Theory , 1st ed., Lecture Notes in Physics, Vol. 190 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1983) pp. IX, 154

1983
[30]

Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10, 285 (1975)

M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10, 285 (1975)

1975
[31]

Bengtsson and K

I. Bengtsson and K. Zyczkowski, Geometry of Quan- tum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2006)

2006
[32]

Georgopoulos, C

K. Georgopoulos, C. Emary, and P. Zuliani, Modeling and simulating the noisy behavior of near-term quantum computers, Phys. Rev. A 104, 062432 (2021)

2021
[33]

P. Zhao, K. Linghu, Z. Li, P. Xu, R. Wang, G. Xue, Y. Jin, and H. Yu, Quantum crosstalk analysis for simul- taneous gate operations on superconducting qubits, PRX Quantum 3, 020301 (2022)

2022
[34]

Winick, J

A. Winick, J. J. Wallman, and J. Emerson, Simulating and mitigating crosstalk, Phys. Rev. Lett. 126, 230502 (2021)

2021
[35]

Zhang, Y

F.-L. Zhang, Y. Jiang, and M.-L. Liang, Speed of disen- tanglement in multiqubit systems under a depolarizing channel, Annals of Physics 333, 136 (2013)

2013
[36]

Siomau and S

M. Siomau and S. Fritzsche, Entanglement dynamics of three-qubit states in noisy channels, The European Phys- ical Journal D 60, 397 (2010)

2010
[37]

J. A. Tropp and R. J. Webber, Randomized algorithms for low-rank matrix approximation: Design, analysis, and applications (2023), arXiv:2306.12418 [math.NA]

2023 arXiv
[38]

Or´ us and G

R. Or´ us and G. Vidal, Infinite time-evolving block deci- mation algorithm beyond unitary evolution, Phys. Rev. B 78, 155117 (2008)

2008
[39]

Vidal, Efficient Classical Simulation of Slightly En- tangled Quantum Computations, Phys

G. Vidal, Efficient Classical Simulation of Slightly En- tangled Quantum Computations, Phys. Rev. Lett. 91, 147902 (2003)

2003
[40]

Y. Sung, L. Ding, J. Braum¨ uller, A. Veps¨ al¨ ainen, B. Kan- nan, M. Kjaergaard, A. Greene, G. O. Samach, C. Mc- Nally, D. Kim, A. Melville, B. M. Niedzielski, M. E. Schwartz, J. L. Yoder, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Realization of High-Fidelity CZ and ZZ...

2021
[41]

Quafu cloud quantum computation platform, https:// quafu.baqis.ac.cn
[42]

Verghese, D

A. Verghese, D. Byron, A. Amann, and E. Popovici, Max-cut problem implementation and analysis on a quantum computer, 2022 33rd Irish Signals and Sys- tems Conference (ISSC) (IEEE, Piscataway, NJ, 2022), pp. 1–6

2022
[43]

M. R. Garey and D. S. Johnson, Computers and In- tractability; A Guide to the Theory of NP-Completeness (W. H. Freeman & Co., USA, 1990)

1990
[44]

M. X. Goemans and D. P. Williamson, Improved approx- imation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42, 1115 (1995)

1995
[45]

Heinz and G

I. Heinz and G. Burkard, Crosstalk analysis for single- qubit and two-qubit gates in spin qubit arrays, Phys. Rev. B 104, 045420 (2021)

2021
[46]

Jozsa, Fidelity for mixed quantum states, Journal of Modern Optics 41, 2315 (1994)

R. Jozsa, Fidelity for mixed quantum states, Journal of Modern Optics 41, 2315 (1994)

1994
[47]

Guo and S

Y. Guo and S. Yang, Quantum error mitigation via ma- trix product operators, PRX Quantum 3, 040313 (2022)

2022

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [14]

K. Noh, L. Jiang, and B. Fefferman, Efficient classical simulation of noisy random quantum circuits in one di- mension, Quantum 4, 318 (2020)

work page 2020

[2] [1]

Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018)

J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018)

2018

[3] [2]

Cheng, X.-H

B. Cheng, X.-H. Deng, X. Gu, Y. He, G. Hu, P. Huang, J. Li, B.-C. Lin, D. Lu, Y. Lu et al., Noisy intermediate- scale quantum computers, Frontiers of Physics 18, 21308 (2023)

work page 2023

[4] [3]

#"#!!! !!! $

(4) B. Noise in Circuit Model Quantum circuits are inherently susceptible to various forms of noise that can significantly affect their perfor- mance. It is crucial to set up models that simplify the representation of quantum noise and help manage this complexity. One such model, the unified noisy circuit model [31], begins with two qubits in the state |0...

work page

[5] [4]

Aaronson, Quantum computing, postselection, and probabilistic polynomial-time, Proc

S. Aaronson, Quantum computing, postselection, and probabilistic polynomial-time, Proc. R. Soc. A 461, 3473 (2005)

work page 2005

[6] [5]

M. J. Bremner, R. Jozsa, and D. J. Shepherd, Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy, Proc. R. Soc. A467, 459 (2011)

work page 2011

[7] [6]

M. J. Bremner, A. Montanaro, and D. J. Shepherd, Average-case complexity versus approximate simulation of commuting quantum computations, Phys. Rev. Lett. 117, 080501 (2016)

2016

[8] [7]

Y. Zhou, E. M. Stoudenmire, and X. Waintal, What lim- its the simulation of quantum computers?, Phys. Rev. X 10, 041038 (2020)

2020