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arXiv:2306.04878 · Physical Review A

Quantum Wasserstein distance between unitary operations

Xinyu Qiu, Lin Chen, Li-Jun Zhao

Published Jul 1, 2024• 1 citation
arXivHTMLPDF

Abstract

Quantifying the effect of noise on unitary operations is an essential task in quantum information processing. We propose the quantum Wasserstein distance between unitary operations, which shows an explanation for quantum circuit complexity and characterizes the local distinguishability of multiqudit operations. We show analytical calculations of the distance between identity and widely used quantum gates including swap, cnot, and other controlled gates. As an application, we estimate the closeness between quantum gates in a circuit and show that the noisy operation simulates the ideal one well when they become close under the distance. Further, we introduce the ${W}_{1}$ error rate by the distance and establish the relation between the ${W}_{1}$ error rate and two practical cost measures of recovery operations in quantum error correction under typical noise scenarios. These applications allow the distance to quantify the effect of noise on unitary operations from the perspective of experiment resources, which cannot be achieved by existing distance.

DOI
10.1103/physreva.110.012412

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Quantifying the effect of noise on unitary operations is an essential task in quantum information processing. We propose the quantum Wasserstein distance between unitary operations, which shows an explanation for quantum circuit complexity and characterizes the local distinguishability of multiqudit operations. We show analytical calculations of the distance between identity and widely used quantum gates including swap, cnot, and other controlled gates. As an application, we estimate the closeness between quantum gates in a circuit and show that the noisy operation simulates the ideal one well when they become close under the distance. Further, we introduce the W1{W}_{1}W1​ error rate by the distance and establish the relation between the W1{W}_{1}W1​ error rate and two practical cost measures of recovery operations in quantum error correction under typical noise scenarios. These applications allow the distance to quantify the effect of noise on unitary operations from the perspective of experiment resources, which cannot be achieved by existing distance.
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Abstract Given a unitary transformation, what is the size of the smallest quantum circuit that implements it? This quantity, known as the quantum circuit complexity, is a fundamental property of quantum evolutions that has widespread applications in many fields, including quantum computation, quantum field theory, and black hole physics. In this paper, we obtain a new lower bound for the quantum circuit complexity in terms of a novel complexity measure that we propose for quantum circuits, which we call the quantum Wasserstein complexity. Our proposed measure is based on the quantum Wasserstein distance of order one (also called the quantum earth mover’s distance), a metric on the space of quantum states. We also prove several fundamental and important properties of our new complexity measure, which stand to be of independent interest. Finally, we show that our new measure also provides a lower bound for the experimental cost of implementing quantum circuits, which implies a quantum limit on converting quantum resources to computational resources. Our results provide novel applications of the quantum Wasserstein distance and pave the way for a deeper understanding of the resources needed to implement a quantum computation.
Despite advances in the development of quantum computers, the practical application of quantum algorithms requiring deep circuit depths or high-fidelity transformations remains outside the current range of the so-called noisy intermediate-scale quantum devices. Now and beyond, quantum circuit compilation (QCC) is a crucial component of any quantum algorithm execution. Besides translating a circuit into hardware-specific gates, it can optimize circuit depth and adapt to noise. Variational quantum circuit compilation (VQCC) optimizes the parameters of an ansatz according to the goal of reproducing a given unitary transformation. In this work, we present a VQCC-objective function called the quantum Wasserstein compilation (QWC) cost function based on the quantum Wasserstein distance of order 1. We show that the QWC cost function upper bounds the average infidelity of two circuits. An estimation method based on measurements of local Pauli-observable is utilized in a generative adversarial network to learn a given quantum circuit. We demonstrate the efficacy of the QWC cost function by compiling hardware efficient ansatz (HEA) as both the target and the ansatz and comparing to cost functions such as the Loschmidt echo test (LET) and the Hilbert-Schmidt test (HST). Finally, our experiments demonstrate that QWC as a cost function is the least affected by barren plateaus when compared to LET and HST for deep enough circuits.
Quantum error correction uses the measurement of syndromes and classical decoding algorithms to estimate the location and type of errors while protecting the encoded quantum bits. Here we consider how prior information and Bayesian updates can play a critical role in improving the performance of QEC in the scenario of a particularly noisy qubit. This allows for leveraging even distance codes, which typically are less valued in QEC, to handle the noisy qubit, changing the power-law scaling of the logical error rate with the baseline physical error rate. A crucial component of this is updating the prior by real time feeding of decoder outputs into a approximate Kalman filter. Thus our approach provides a bootstrap to the actual error rates. We show this via simulation of the full closed-loop system: starting from uniform priors, the update procedure gradually learns site-specific error rates, enabling the decoder to outperform a fixed-prior baseline. In turn, we show that this enables in situ calibration of unitary operations via injection of gate set tomography operations with only moderate overhead in the more typical scenario of low noise qubits.