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Modelling and Simulating the Noisy Behaviour of Near-term Quantum Computers

Konstantinos Georgopoulos, Clive Emary, Paolo Zuliani

arXiv:2101.02109v4quant-ph

TL;DR

Noise obstructs long quantum computations, and the paper addresses the challenge of modelling that noise in quantum computers. It combines channels for three error groups into an architecture-aware model and optimizes selected parameters. The model more accurately approximates IBM hardware than other models, with optimization yielding up to 84% better approximation.

  • Problem

    Noise is a central obstacle to building large-scale quantum computers and executing long quantum computations.

    Source: §I · p. 1

  • Method

    The paper models gate infidelities, SPAM errors, and thermal decoherence and dephasing with quantum channels combined into an architecture-aware noise model, then optimizes a subset of its parameters.

    Source: §I · p. 1

  • Results

    The unified model more accurately approximates IBM quantum-computer evolution than other models, and parameter optimization yields up to 84% better approximation.

    Source: §I · p. 1

  • Takeaways & Limitations

    The model can help characterize quantum-computer noise and support the design or testing of error-correction methods and calibration.

    Source: §VI · p. 12

    SPAM errors are modelled exclusively as hardware errors, primarily Pauli-X, although other preparation and measurement errors may occur.

    Source: §VI · p. 11

Abstract

from arXiv · show

Noise dominates every aspect of near-term quantum computers, rendering it exceedingly difficult to carry out even small computations. In this paper we are concerned with the modelling of noise in Noisy Intermediate-Scale Quantum (NISQ) computers. We focus on three error groups that represent the main sources of noise during a computation and present quantum channels that model each source. We engineer a noise model that combines all three noise channels and simulates the evolution of the quantum computer using its calibrated error rates. We run various experiments of our model, showcasing its behaviour compared to other noise models and an IBM quantum computer. We find that our model provides a better approximation of the quantum computer's behaviour than the other models. Following this, we use a genetic algorithm to optimize the parameters used by our noise model, bringing the behaviour of the model even closer to the quantum computer. Finally, a comparison between the pre and postoptimization parameters reveals that, according to our model, certain operations can be more or less erroneous than the hardware-calibrated parameters show.

I. INTRODUCTION

The paper models three major quantum-computing error sources with separate channels combined into an architecture-aware noise model, which more accurately approximates IBM hardware than competing models. Optimizing selected parameters further improves the approximation, while the introduction also motivates noise modelling as relevant to calibration and error correction.

I. INTRODUCTION

  • Noise obstructs large-scale and long quantum computations, while identifying system-specific noise can inform more efficient calibration and error correction.Error-correction methods optimized for a system’s noise can outperform generic ones.Source: §I · p. 1
  • The unified model more accurately approximates an IBM quantum computer’s evolution than other models, and parameter optimization yields up to 84% better approximation.The model combines gate infidelities, SPAM errors, and thermal decoherence and dephasing; it uses hardware-calibrated parameters and is architecture-aware.Source: §I · p. 1

II. QUANTUM NOISE CHANNELS

  • The model treats gate infidelities, state preparation and measurement errors, and thermal decoherence and dephasing as distinct quantum-channel error groups.It simulates noisy evolution using the computer’s calibrated error rates.Source: §I · p. 1, §II · p. 2

I. INTRODUCTION

  • The model is not limited to IBM quantum computers and can be attached to QASM-based implementations.The implementation uses IBM’s Qiskit development kit.Source: §I · pp. 1–2

A. Error Group 1: Depolarizing Channel

  • A depolarizing Pauli channel models gate infidelities by applying equally probable X, Y, or Z errors, but non-unital environmental noise requires a more complicated approach.Other forms of decoherence, including electromagnetic and gravitational effects, are not considered.Source: §I · p. 1, A. Error Group 1: Depolarizing Channel · p. 2

B. Error Group 2: State Preparation and Measurement (SPAM) Channel

The SPAM channel treats preparation and measurement errors as Pauli-X flips, while the third error group models relaxation and dephasing over time. The thermal model assumes independent per-qubit noise and omits rare thermal excitation under its stated conditions.

B. Error Group 2: State Preparation and Measurement (SPAM) Channel

  • Measurement errors use a Pauli-X channel with probability p2, while state-preparation errors similarly invert the desired state with probability p′2.The paper groups preparation and measurement because both are modelled in the same way, while keeping them separate from the main circuit execution.Source: B. Error Group 2: State Preparation… · p. 2

C. Error Group 3: Thermal Decoherence and Dephasing Channel

  • The thermal decoherence and dephasing group models relaxation or excitation over time alongside the decay of qubit coherence.The model assumes these effects occur independently for each qubit; it omits thermal excitation as rare under the stated temperature and hardware example.Source: C. Error Group 3: Thermal Decoherence… · p. 3
  • For each gate, the model uses its execution time and each qubit’s T1(q) and T2(q) times to calculate relaxation and dephasing probabilities.The reset probability is defined as 1 − pT1(q).Source: C. Error Group 3: Thermal Decoherence… · p. 3
  • When T2(q) is greater than T1(q) but no greater than 2T1(q), the implementation uses a Choi-matrix representation for the channel.The Choi-matrix evolution is converted to an operator-sum representation using eigenvalues when applicable or singular value decomposition otherwise.Source: C. Error Group 3: Thermal Decoherence… · pp. 3–4

III. UNIFIED MODEL FOR QUANTUM NOISE

After defining the individual noise channels, the paper introduces a unified quantum noise model.

III. UNIFIED MODEL FOR QUANTUM NOISE

  • The unified-model section follows the individual definitions of the three quantum noise channels.Source: §III · p. 4

A. Quantum Noise Parameters

The model’s channel parameters generally come from quantum-computer calibration. Gate and SPAM rates vary across operation types and qubits, and the parameter inventory is organized by error group.

A. Quantum Noise Parameters

  • The quantum noise channels use parameters that are usually obtained through calibration, and IBM quantum-computer calibration parameters are publicly available.Source: A. Quantum Noise Parameters · p. 4
  • Gate-error parameters give the probability of an erroneous gate outcome and vary by gate type and by the qubits to which each gate is applied.The table describes parameter types and counts for each error group, with counts depending on n, m, s, and r.Source: A. Quantum Noise Parameters · p. 4, Table I · p. 4
  • SPAM parameters represent the probabilities of erroneous state preparation or measurement, with different rates for each qubit.Source: A. Quantum Noise Parameters · p. 4

B. Constructing the Unified Model

The unified model combines architecture-aware noise channels for gate infidelities, SPAM, and thermal relaxation and dephasing. It applies these channels at hardware-relevant points in the circuit, accounting for qubit connectivity and gate properties.

B. Constructing the Unified Model

  • The model accounts for qubit connectivity and each qubit’s properties, as well as the execution times and error rates of participating gates.Gate execution times are assumed static for each quantum computer.Source: B. Constructing the Unified Model · pp. 4–5
  • Depolarizing errors follow single-qubit gates using single-qubit error rates and two-qubit gates using rates informed by hardware connectivity.The model applies the SPAM channel after state preparation, if present, and before measurement.Source: B. Constructing the Unified Model · p. 4
  • The relaxation and dephasing channel acts on each qubit after every gate according to that qubit’s relaxation and dephasing times and the gate duration.The model applies this channel after the depolarizing channel.Source: B. Constructing the Unified Model · p. 5, Figure 1 · p. 5
  • The unified model composes the independent noise channels with circuit gates to simulate the evolution from an initial state.The channel definition can be expanded to higher-dimensional operators, including two-qubit operations.Source: B. Constructing the Unified Model · p. 5
  • For two-qubit gates, the depolarizing channel affects only the target qubit; the control qubit acts as a driver and is unaffected by the gate or its noise.The channels are applied independently to each qubit, and the identity is treated as a virtual gate with zero execution time.Source: B. Constructing the Unified Model · p. 5, Figure 2 · p. 5

IV. SIMULATING NOISE IN QUANTUM COMPUTERS

The paper simulates quantum walks on a quantum computer and with noise models, using the Hellinger distance to compare their output distributions. Its preliminary methods describe the walk’s noise sensitivity, circuit implementation, and the hardware parameters used by the architecture-aware model.

IV. SIMULATING NOISE IN QUANTUM COMPUTERS

  • The unified model is implemented in Python, and circuits are executed on IBMQ Qiskit simulators and the 15-qubit Melbourne computer.The authors state that the model can also be applied to other architectures using QASM or QASM-type low-level circuit implementations.Source: §IV · p. 5

A. Preliminary Methods

  • Discrete-time quantum walks are used because their predictable behavior is susceptible to noise, which can violate their modular parity relationship.The walk circuit uses inverter gates, with gate count increasing as O(log2 N) with state-space size N.Source: A. Preliminary Methods · p. 6
  • The Hellinger distance compares simulated and computer-generated probability distributions, ranges from 0 to 1, and equals 0 for identical distributions.It does not require the distributions to have the same support.Source: A. Preliminary Methods · p. 6
  • The architecture-aware model uses per-qubit error rates and decoherence times, plus pairwise information reflecting qubit connectivity.Table II reports averages across the 15-qubit Melbourne machine for the experiment date.Source: A. Preliminary Methods · p. 6, Table II · p. 6

B. Experiments and Results with the Calibrated Parameters

With calibrated parameters, the unified noise model most closely approximates the quantum computer among the evaluated models, though model accuracy declines as circuit size grows. For walks larger than N = 8, the computer’s output distributions are closer to uniform amid excessive noise.

B. Experiments and Results with the Calibrated Parameters

  • QiskitCM’s use of averaged rather than per-qubit noise parameters is associated with a larger deviation from the quantum computer than UNM.The paper describes DSPAM and TRM as simpler implementations of separate error groups, and SDM as a simple depolarizing model that does not account for qubit connectivity.Source: B. Experiments and Results with the… · pp. 6–7
  • The experiments compare the unified model with QiskitCM, DSPAM, TRM, and SDM, using 100,000 runs on the computer and in each simulation.QiskitCM averages noise parameters across qubits rather than accounting for each separately, and the paper reports a larger deviation from the computer than for UNM.Source: B. Experiments and Results with the… · pp. 6–7
  • The unified noise model performs best overall, followed closely by QiskitCM, while all models’ distances from the computer increase with qubit count.For a three-qubit walk, the unified model has the smallest Hellinger distance from the computer, 0.12749.Source: B. Experiments and Results with the… · p. 7
  • For walks with N > 8, the quantum computer produces distributions closer to uniform, which the authors attribute to excessive QPU noise.The authors include these results as driven by noise effects and intensity within the QPU.Source: B. Experiments and Results with the… · p. 7

C. Unified Noise Model vs Gate Set Tomography

The unified noise model simulates noisy circuit evolution, whereas gate set tomography characterizes gate operations and requires the gates to be specified beforehand. In the QEC context, the paper presents combined noise models as useful and notes that expanding their channels may improve fidelity.

C. Unified Noise Model vs Gate Set Tomography

  • UNM simulates a circuit’s noisy evolution at runtime across gate, SPAM, and decoherence effects, while GST characterizes gate operations after the gates are specified.GST works well with small systems, whereas UNM aims to scale with circuit size, subject to classical simulation limits.Source: C. Unified Noise Model vs Gate Set… · p. 7
  • UNM depends on calibrated noise parameters, unlike GST’s calibration-free reconstruction, making calibration essential to the unified model’s success.This dependence motivates the paper’s noise-parameter optimization technique.Source: C. Unified Noise Model vs Gate Set… · p. 7

D. Noise Modelling in Quantum Error Correction

  • The paper situates UNM alongside QEC noise models that distinguish gate errors, environmental decoherence, and loss, leakage, measurement, and initialization errors.It notes that combining models can address the need for more specific decoherence models in settings such as superconducting systems.Source: D. Noise Modelling in Quantum Error… · pp. 7–8
  • Adding further systematic or environmental noise types, such as Clifford errors or electromagnetic noise, may make UNM more accurate.The paper describes UNM as combining major error sources relevant to quantum error mitigation.Source: D. Noise Modelling in Quantum Error… · p. 8

V. OPTIMIZING THE QUANTUM NOISE PARAMETERS

The paper optimizes a subset of the unified noise model’s parameters with a genetic algorithm to bring simulated quantum-walk distributions closer to the quantum computer. Optimization improves the two reported small-walk comparisons, while larger state spaces make the search slower and less efficient under a fixed generation count.

V. OPTIMIZING THE QUANTUM NOISE PARAMETERS

  • The search covers depolarizing-Pauli and SPAM parameters, counting gate types, measured qubits, and prepared qubits; the quantum-walk gate count groups similar single-qubit error rates.For the reported walks, the count is based on distinct gate types and connected two-qubit gates, not how often a gate is used.Source: §V · p. 9

A. Parameter Optimization

  • Relaxation and dephasing parameters T1(q) and T2(q) are excluded because qubit-wise search increases parameter count and runtime; reported gains were similar or lower when they were included.For the two-qubit walk, gains were 84.85% without versus 85.48% with decoherence optimization; for the three-qubit walk, they were 72.44% versus 64.57%, respectively.Source: §V · p. 9, A. Parameter Optimization · pp. 10–11
  • The optimization uses a genetic algorithm that iteratively simulates candidate parameters and retains those that bring the simulated distribution closer to the computer’s.The parameters are supplied to the unified noise model before simulation; unlike QiskitCM’s automatic hardware-parameter loading, this can be controlled in the model’s implementation.Source: §V · p. 8, A. Parameter Optimization · p. 9
  • 84.85%: For the two-qubit walk, optimization reduced Hellinger distance from ∼0.033 to ∼0.005 relative to the quantum computer.The comparison followed one optimization routine with 50 generations.Source: A. Parameter Optimization · p. 9, Figure 4 · p. 10
  • 57%: For the N = 8 walk, optimization reduced Hellinger distance from ∼0.127 to ∼0.054 relative to the quantum computer.This comparison also used 50 generations.Source: A. Parameter Optimization · p. 9, Figure 5 · p. 10
  • As state space grows, ancillary qubits increase the parameter count and slow the genetic-algorithm routine; with 50 generations, postoptimization efficiency declines, although more generations can improve approximation.Table IV reports averages over three optimization routines for 50 and 100 generations.Source: A. Parameter Optimization · pp. 9–10, Table IV · p. 10

B. Noise Parameters Analysis

The unified model’s optimized parameters improve its approximation of the computer’s noisy evolution, while parameter comparisons suggest calibration errors vary by operation and qubit. These conclusions are qualified by optimization and model-scope limitations.

B. Noise Parameters Analysis

  • For N = 4, single-qubit operations and measurements on qubits 0 and 1 are noisier than calibration suggests, while qubits 2 and 3 and all two-qubit operations are less noisy.Table V reports the pre- and postoptimization parameters for the N = 4 state-space system.Source: B. Noise Parameters Analysis · p. 11, Table V · p. 12
  • In larger systems, most single-qubit operations and measurements tend to be noisier than calibrated, while two-qubit operations tend to be less noisy.The analysis averages three optimization routines; longer experiments exceeding T1 and T2 make concrete conclusions difficult, and further runs could change the pattern.Source: B. Noise Parameters Analysis · p. 11

VI. CONCLUSIONS AND FUTURE WORK

  • The optimized unified model can be more than 84% closer to the computer’s noisy evolution than the unoptimized model.The authors also report that the model’s fitted error rates can differ from hardware calibrations.Source: §VI · p. 11
  • The unified model uses a highly Markovian approach, performs satisfactorily especially on small systems, and leaves non-Markovian noise as a proposed extension.The authors suggest investigating its pros and cons within the framework.Source: §VI · p. 11
  • The model limits SPAM errors to hardware errors with Pauli-X as the most common error, although other Pauli errors and operation-time decoherence could also occur.The authors suggest these additional noise types as possible extensions and say including them could improve precision.Source: §VI · p. 11
  • The authors identify two optimization contributions: a technique they say had not previously been carried out to their knowledge, and a unified model that permits optimization unlike the compared IBMQ models.They also suggest more genetic-algorithm iterations or other optimization techniques as possible routes to more accurate parameters.Source: §VI · pp. 11–12
  • The authors say the model can help characterize quantum-computer noise and support the design or testing of error-correction methods and calibration techniques.They also describe it as a way to help minimize noise in near-term quantum computers.Source: §VI · p. 12

Appendix A: Thermal Relaxation and Dephasing Model

The appendix describes thermal relaxation and dephasing channels across parameter regimes, deriving Kraus operators from Choi-matrix representations when needed. The resulting channel construction can fail its required completely positive trace-preserving condition when left and right operators differ.

Appendix A: Thermal Relaxation and Dephasing Model

  • For T2 ≤ T1, the appendix considers thermal excitation at processor temperature Θ ≥ 0; for 2T1 ≥ T2 > T1, it uses a Choi-matrix representation.The channel construction uses the previously defined T1, T2, Tg, pT1, pT2, and reset probability.Source: Appendix A · p. 12
  • For the described noise case, the channel includes phase flips, resets to |0⟩ or |1⟩, and identity operations, represented by corresponding Kraus operators.The reset to |1⟩ represents spontaneous excitation; the identity represents no change to the state.Source: Appendix A · p. 12
  • When a Choi matrix is not Hermitian or has negative eigenvalues, the method applies singular value decomposition to obtain left and right Kraus-operator sets.The singular values and left and right singular vectors are used to construct the two sets.Source: Appendix A · p. 13
  • If the left and right Kraus operators differ, they do not represent a completely positive trace-preserving map, triggering an error in the thermal relaxation model.Source: Appendix A · p. 13

Appendix B: Circuit for Discrete-time Quantum Walk

The appendix describes a discrete-time quantum walk circuit built from increment and decrement functions, and a separate optimization experiment that includes relaxation and dephasing parameters. In the reported experiment, optimization performs better when those decoherence parameters are excluded.

Appendix B: Circuit for Discrete-time Quantum Walk

  • The quantum-walk circuit uses generalized controlled gates and sequences increment and decrement functions to increase or decrease the quantum-register state.Figure 7 depicts one walk step and the increment/decrement circuits; filled and empty control circles denote controls on |1⟩ and |0⟩, respectively.Source: Appendix B · p. 13, Figure 7 · p. 13

Appendix C: Genetic Algorithm Optimization Including the Decoherence Parameters

  • For N = 4, the decoherence-inclusive optimization has 17 parameters; for N = 8, it has 26, including hardware/SPAM errors and per-qubit relaxation and dephasing times.The reported genetic-algorithm routine uses 100 generations.Source: Appendix C · p. 13
  • Optimization performs better in both tested state-space cases when T1(q) and T2(q) are excluded, partly because more parameters reduce the potential optimum search space and the model imposes T2(q) ≤ 2T1(q).The comparison concerns optimization routines with and without decoherence parameters.Source: Appendix C · p. 14
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