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Mitigating depolarizing noise on quantum computers with noise-estimation circuits
Miroslav Urbanek, Benjamin Nachman, Vincent R. Pascuzzi, Andre He, Christian W. Bauer, Wibe A. de Jong
TL;DR
Noise limits current quantum computers, while depolarizing noise can describe average noise in large circuits. The paper estimates that noise with a structurally similar circuit, corrects the target output, and combines the method with randomized compiling and zero-noise extrapolation. The combined approach produces results close to exact results, including for circuits with hundreds of CNOT gates.
Problem
Noise limits the utility of current non-fault-tolerant quantum computers, motivating methods to mitigate errors in their outputs.
Method
The method constructs an estimation circuit similar to the target, estimates the depolarizing noise rate, and uses it to correct the target output alongside randomized compiling and zero-noise extrapolation.
Results
The combined approach produces results very close to exact results for a Heisenberg-model simulation, including circuits containing hundreds of CNOT gates.
Takeaways & Limitations
The method works well in combination with readout-error correction and zero-noise extrapolation on test circuits containing hundreds of CNOT gates.
Abstract
from arXiv · showhide
A significant problem for current quantum computers is noise. While there are many distinct noise channels, the depolarizing noise model often appropriately describes average noise for large circuits involving many qubits and gates. We present a method to mitigate the depolarizing noise by first estimating its rate with a noise-estimation circuit and then correcting the output of the target circuit using the estimated rate. The method is experimentally validated on the simulation of the Heisenberg model. We find that our approach in combination with readout-error correction, randomized compiling, and zero-noise extrapolation produces results close to exact results even for circuits containing hundreds of CNOT gates.
I. INTRODUCTION
NISQ quantum computers remain limited by noise and errors, motivating mitigation methods. This work introduces estimation-circuit mitigation and combines it with other techniques to approach exact results on Heisenberg-model simulations.
- Motivation: Noise and errors limit the utility of current and near-term quantum computers that are not fault-tolerant.The passage notes that error rates remain too large for relevant scientific problems on existing devices.
- Existing mitigation: Readout errors produce incorrect measured qubit values and can be mitigated with readout-error correction.Several correction methods with varying sophistication have been developed.
- Error sources: Gate errors include coherent and incoherent components, with coherent errors producing systematic output bias.Coherent errors arise from small control-parameter miscalibrations, whereas incoherent errors involve randomly varying controls or environmental entanglement.
- Existing mitigation: Zero-noise extrapolation executes circuits at varying noise levels and extrapolates measured outputs to the zero-noise limit.The paper places this technique among several existing error-mitigation approaches.
- Contribution: The proposed method estimates depolarizing noise with an estimation circuit and uses the measured rate to correct the target circuit's output.The authors experimentally combine it with readout-error correction, randomized compiling, and zero-noise extrapolation.
- Contribution: The full mitigation combination produces results very close to exact results in a simulation of the Heisenberg model.The paper then presents the method and evaluates improvements on this test case.
II. METHODS
The method models incoherent noise as depolarizing noise, estimates its rate using a structurally similar circuit, and corrects noisy observables. Randomized compiling and zero-noise extrapolation extend the approach to coherent errors and experimentally usable mitigation.
- Depolarizing-noise model: The depolarizing channel mixes a density matrix with the maximally mixed state using probabilistic error rate p.The rate depends on both the device and the circuit.
- Observable correction: Under depolarizing noise, non-identity Pauli components of an observable's expectation value are scaled by 1−p.Identity-only Pauli strings retain expectation value 1.
- Observable correction: An observable is decomposed into an identity contribution and a traceless component before correcting its noisy expectation value using the estimated noise rate.The identity term is a known constant shift and is set to c = 0 in the subsequent treatment.
- Estimation circuits: The estimation circuit removes single-qubit gates, retains CNOT gates, and estimates 1−p from the probability of measuring the all-zero state.This relies on ideal CNOT gates leaving the initial zero state unchanged and assumes similar noise rates for target and estimation circuits.
- Estimation circuits: Estimation circuits may preserve some single-qubit gates and add inverse random-rotation layers to remain structurally similar and improve robustness.The circuit only needs a known output sensitive to noise and a structure similar to the target circuit.
- Comparison with related methods: Unlike near-Clifford mitigation, this method does not select biased random circuits; it removes single-qubit gates to obtain a trivially simulable biased circuit.The near-Clifford approach uses machine learning to identify suitable circuits.
- Randomized compiling: Randomized compiling inserts random identity or Pauli single-qubit gates around CNOT layers while preserving each CNOT's overall action.This converts coherent errors into incoherent errors so the depolarizing model is more applicable.
- Zero-noise extrapolation: Zero-noise extrapolation replaces each CNOT with one, three, or five consecutive CNOTs and uses quadratic extrapolation to the zero-CNOT limit.The method assumes CNOT gates are the dominant error source.
III. EXPERIMENT
The experiment simulates local magnetization dynamics in an XX chain using Trotterized circuits, then combines estimation-circuit mitigation with randomized compiling, readout-error correction, and zero-noise extrapolation. Noise is artificially varied by replacing each CNOT with one, three, or five CNOTs, enabling quadratic extrapolation to the zero-noise limit.
- System and circuit: The XX-chain experiment measures the local magnetization of the last spin after second-order Trotter–Suzuki time evolution.The model uses three alternating layers of two-qubit circuits, with each two-qubit exponential decomposed into two CNOT gates and single-qubit gates.
- System and circuit: The six-qubit implementation uses 14 CNOT gates per time step, reaching 210 CNOT gates after 15 time steps.The experiment was run on IBM Q Paris with 8192 shots per circuit.
- Noise estimation: Estimation circuits remove target-circuit single-qubit gates, add random single-qubit layers and their inverses, and retain the CNOT structure.Versions with one, three, and five CNOT gates replacing each original CNOT were executed with 448 randomized instances.
- Mitigation: The measured depolarizing rate is used to correct expectation values before zero-noise extrapolation.The extrapolation uses n = 1, 3, and 5 and a quadratic fit to estimate the n = 0 value.
- Mitigation: Quadratic extrapolation works well after estimation-circuit mitigation because the residual noise dependence is close to an exponential near n = 0.Direct quadratic extrapolation of target-circuit values performs poorly when the noise exponent is large.
IV. CONCLUSION
The paper concludes that depolarizing noise can be mitigated by estimating its rate with a structurally related circuit and correcting the target output. Combined with randomized compiling, readout-error correction, and zero-noise extrapolation, the method works well on circuits containing hundreds of CNOT gates and is scalable with sufficient randomized sampling.
- Method: The method constructs an estimation circuit to estimate the noise rate and uses that estimate to correct a given circuit’s output.The approach targets errors described by the depolarizing noise model.
- Method: Randomized compiling is crucial because it enables gate errors to be modeled as incoherent depolarizing noise.
- Results: The combined method works well, especially with readout-error correction and zero-noise extrapolation, on test circuits containing hundreds of CNOT gates.
- Scope: The approach is scalable to any number of qubits and gates when enough randomized samples are collected to achieve low uncertainty.
Appendix A: Zero-noise extrapolation
The appendix describes artificial noise scaling by replacing each CNOT with one, three, or five consecutive CNOT gates. These replacements preserve the ideal noiseless operation while increasing the implemented noise for mitigation and extrapolation.
- Noise scaling: Each CNOT gate is replaced by one, three, or five consecutive CNOT gates to artificially increase noise.The modified gates are equivalent to a single CNOT on a noiseless quantum computer.
- Noise scaling: Both estimation and target circuits are modified before mitigation uses their measured outputs.