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Mitigating depolarizing noise on quantum computers with noise-estimation circuits

Miroslav Urbánek, Benjamin Philip Nachman, Vincent Richard Pascuzzi, Andre He, C. Bauer, Wibe Albert de Jong

Published Mar 15, 2021• 19 citations
OpenAlex

Abstract

A significant problem for current quantum computers is noise. While there are\nmany distinct noise channels, the depolarizing noise model often appropriately\ndescribes average noise for large circuits involving many qubits and gates. We\npresent a method to mitigate the depolarizing noise by first estimating its\nrate with a noise-estimation circuit and then correcting the output of the\ntarget circuit using the estimated rate. The method is experimentally validated\non the simulation of the Heisenberg model. We find that our approach in\ncombination with readout-error correction, randomized compiling, and zero-noise\nextrapolation produces results close to exact results even for circuits\ncontaining hundreds of CNOT gates.

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A significant problem for current quantum computers is noise. While there are\nmany distinct noise channels, the depolarizing noise model often appropriately\ndescribes average noise for large circuits involving many qubits and gates. We\npresent a method to mitigate the depolarizing noise by first estimating its\nrate with a noise-estimation circuit and then correcting the output of the\ntarget circuit using the estimated rate. The method is experimentally validated\non the simulation of the Heisenberg model. We find that our approach in\ncombination with readout-error correction, randomized compiling, and zero-noise\nextrapolation produces results close to exact results even for circuits\ncontaining hundreds of CNOT gates.
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 a simulation of the Heisenberg model. We find that our approach in combination with readout-error correction, randomized compiling, and zero-noise extrapolation produces close to exact results even for circuits containing hundreds of CNOT gates. We also show analytically that zero-noise extrapolation is improved when it is applied to the output of our method.
Quantum computers have enabled solving problems beyond the current machines' capabilities. However, this requires handling noise arising from unwanted interactions in these systems. Several protocols have been proposed to address efficient and accurate quantum noise profiling and mitigation. In this work, we propose a novel protocol that efficiently estimates the average output of a noisy quantum device to be used for quantum noise mitigation. The multi-qubit system average behavior is approximated as a special form of a Pauli Channel where Clifford gates are used to estimate the average output for circuits of different depths. The characterized Pauli channel error rates, and state preparation and measurement errors are then used to construct the outputs for different depths thereby eliminating the need for large simulations and enabling efficient mitigation. We demonstrate the efficiency of the proposed protocol on four IBM Q 5-qubit quantum devices. Our method demonstrates improved accuracy with efficient noise characterization. We report up to 88% and 69% improvement for the proposed approach compared to the unmitigated, and pure measurement error mitigation approaches, respectively.
We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary kkk-qubit gates each of whose input wires is subject to depolarizing noise of strength ppp, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p>1−Θ(1/k)p>1-\Theta(1/\sqrt{k})p>1−Θ(1/k​), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k=2k=2k=2, our bound is p>35.7%p>35.7\%p>35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes 29.3%29.3\%29.3%. These bounds on ppp are numerically better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, in which the effects of depolarizing noise are very easy to describe.
Quantum circuit ensembles that have the properties of unitary k-designs represent applications where there is no obvious bias toward any particular Pauli support, as is the case in simulating systems exhibiting ''quantum chaos,'' which range from quantum dynamics near black holes to gapless spin fluid analysis. However, noisy hardware makes quantum circuits prone to a myriad of error sources, of which depolarizing and coherent error can be particularly destructive. To combat depolarizing error, popular techniques typically involve circuit or gate folding, which can be time-intensive procedures due to increased circuit depth and shot overhead. Other tensor-network-based mitigation techniques suffer from intractability in high-entanglement regimes. In this work, we leverage the structure of unitary k-design Pauli support distributions by introducing a technique we name ''circuit balancing,'' along with gate benchmarking data, in order to estimate circuit-wide depolarization. We describe how to invert the diagnosed circuit depolarization even in the presence of coherent error, via Pauli twirling. We provide asymptotics to estimate the number of twirls needed to maintain a desired output fidelity. We test our method numerically in a variety of simulation settings and find that it can significantly reduce average random circuit infidelity. Further, we employ our methods to find significant infidelity reductions when running a random circuit ensemble on a contemporary superconducting quantum computer, IBM Fez. Overall, we show that the method effectively reduces gate-based error for unitary k-designs without incurring any two-qubit gate overhead.
In the noisy intermediate-scale quantum (NISQ) era, quantum error mitigation (QEM) is essential for producing reliable outputs from quantum circuits. We present a statistical signal processing approach to QEM that estimates the most likely noiseless outputs from noisy quantum measurements. Our model assumes that circuit depth is sufficient for depolarizing noise, producing corrupted observations that resemble a uniform distribution alongside classical bit-flip errors from readout. Our method consists of two steps: a filtering stage that discards uninformative depolarizing noise and an expectation-maximization (EM) algorithm that computes a maximum likelihood (ML) estimate over the remaining data. We demonstrate the effectiveness of this approach on small-qubit systems using circuit simulations in Qiskit and IBM quantum processing unit (QPU) data, and compare its performance to contemporary statistical QEM techniques. We also show that our method scales to larger qubit counts using synthetically generated data consistent with our noise model. These results suggest that principled statistical methods can offer scalable and interpretable solutions for quantum error mitigation in realistic NISQ settings.
Quantum computers in the NISQ era are prone to noise. A range of quantum error mitigation techniques has been proposed to address this issue. Zero-noise extrapolation (ZNE) stands out as a promising one. ZNE involves increasing the noise levels in a circuit and then using extrapolation to infer the zero noise case from the noisy results obtained. This paper presents a novel ZNE approach that does not require circuit folding or noise scaling to mitigate depolarizing and/or decoherence noise. To mitigate depolarizing noise, we propose leveraging the extreme/infinite noisy case, which allows us to avoid circuit folding. Specifically, the circuit output with extreme noise becomes the maximally mixed state. We show that using circuit-reliability metrics, simple linear extrapolation can effectively mitigate depolarizing noise. With decoherence noise, different states decay into the all-zero state at a rate that depends on the number of excited states and time. Therefore, we propose a state- and latency-aware exponential extrapolation that does not involve folding or scaling. When dealing with a quantum system affected by both decoherence and depolarizing noise, we propose to use our two mitigation techniques in sequence: first applying decoherence error mitigation, followed by depolarizing error mitigation. A common limitation of ZNE schemes is that if the circuit of interest suffers from high noise, scaling-up noise levels could not provide useful data for extrapolation. We propose using circuit-cut techniques to break a large quantum circuit into smaller sub-circuits to overcome this limitation. This way, the noise levels of the sub-circuits are lower than the original circuit, and ZNE can become more effective in mitigating their noises.