Establishing trust in quantum computations

TL;DR

This work addresses the challenge of verifying quantum computations on noisy hardware by introducing mirror-circuit fidelity estimation (MCFE). MCFE builds motion-reversal circuits from a target circuit and uses local randomized states plus randomized compilation to transform errors into stochastic Pauli channels, enabling an efficiently computable fidelity surrogate $\chi_F(c)$ that satisfies $\chi_F(c) = F(c) + \mathcal{O}(\Delta_1,\Delta_2)$ under physically reasonable assumptions. The method achieves a fidelity assessment with classical resources that scale linearly with circuit size and do not depend on the number of qubits, demonstrated via simulations of QAOA on up to 100 qubits across several error models. The results show $\hat{\chi}_F(c)$ accurately tracks the true fidelity for common noise types (stochastic Pauli and mixed errors) and thus provide a practical pathway to trust quantum computations in the NISQ era and beyond. The approach promises scalable verification for fault-tolerant quantum circuits and supports credible claims of quantum advantage by quantifying algorithmic execution fidelity $F(c)$ without full tomography.

Abstract

Quantum computing hardware has grown sufficiently complex that it often can no longer be simulated by classical computers, but its computational power remains limited by errors. These errors corrupt the results of quantum algorithms, and it is no longer always feasible to use classical simulations to directly check the correctness of quantum computations. Without practical methods for quantifying the accuracy with which a quantum algorithm has been executed, it is difficult to establish trust in the results of a quantum computation. Here we solve this problem, by introducing a simple and efficient technique for measuring the fidelity with which an as-built quantum computer can execute an algorithm. Our technique converts the algorithm's quantum circuits into a set of closely related ``mirror circuits'' whose success rates can be efficiently measured. It enables measuring the fidelity of quantum algorithm executions both in the near-term, with algorithms run on hundreds or thousands of physical qubits, and into the future, with algorithms run on logical qubits protected by quantum error correction.

Figures (3)

• Figure 1: Establishing trust in quantum computations by estimating circuit execution fidelity.A. A quantum algorithm for gate-based quantum computers solves a computational problem by running one or more quantum circuits on a quantum computer, to sample from some probability distributions $P_{\textrm{ideal}}$. Real-world quantum computers implement circuits imperfectly, resulting in circuit outputs (bit strings) sampled from distributions $P$ that differ from $P_{\textrm{ideal}}$, causing inaccuracy in the algorithm's solution. Quantifying the solution inaccuracy caused by these imperfections is difficult, because, for example, computing $P_{\textrm{ideal}}$ is generally infeasible using even the most powerful supercomputers. B. Our technique estimates the fidelity $F(c)$ with which an as-built quantum computer can execute some target circuit $c$ (green box), using motion reversal circuits built from $c$ and the four reference (sub)circuits shown here. C. The three motion reversal circuits we use to estimate fidelity, $M_i(c)$, selectively use randomized compilation to isolate $F(c)$ without requiring that $c$ is randomly compiled. These motion reversal circuits are designed so that their mean adjusted success probabilities $S(\cdot)$ [see \ref{['eq:S']}] are approximately equal to the product of the fidelities of their constituent subcircuits. This means that the simple function of their mean adjusted success probabilities shown in D can be used to estimate $F(c)$.
• Figure 2: Applying our protocol to QAOA circuits using simulated data.A. Parameterized QAOA circuits for approximately solving the MaxCut problem on a weighted $n$-vertex graph consist of alternating applications of unitaries generated by two Hamiltonians: $H_C$, which encodes the graph, and $H_D$, which is a driver Hamiltonian that does not commute with $H_C$Farhi2014-bt. These unitaries are compiled into single-qubit gates and CNOTs. We randomly selected 300 distinct QAOA circuits, on up to $n=8$ qubits and with up to $p=10$ algorithmic layers. We applied our protocol to each circuit, with simulated data from four different families of error models. The variational parameters were randomly assigned for each circuit. B. Each circuit's true process fidelity [$F(c)$], under the selected error model, versus the process fidelity predicted by our protocol [$\hat{\chi}_F(c)$], calculated by sampling 1000 mirror circuits of each type (see Fig. \ref{['fig:circuits']}C) and simulating them under that error model. The four error models consist of stochastic Pauli errors on all gates, stochastic Pauli errors and Hamiltonian errors on all gates, Hamiltonian errors on all gates, and Hamiltonian errors on only two-qubit gates. The rates of the errors in each error model are randomly sampled for each QAOA circuit $c$. Each $\hat{\chi}_F(c)$ is shown with error bars (1 standard deviation, calculated using a nonparametetric boostrap) that, in most cases, are smaller than the data points.
• Figure 3: Measuring the fidelity of 100-qubit circuits. We simulated applying our technique to 100-qubit circuits, demonstrating that it works in the $n=\mathcal{O}(100)$ regime of potential quantum advantage, using only 1,200 circuits and a total of 216,000 samples for each $c$. These simulations use Clifford circuits subject to stochastic Pauli errors so that we can efficiently generate simulated data. For every selected circuit and error model (details in text), we find that the estimate of the fidelity obtained from our protocol [$\hat{\chi}_F(c)$] is a good approximation to the true fidelity [$F(c)$]. Each $\hat{\chi}_F(c)$ is shown with error bars (1 standard deviation, calculated using a nonparametetric boostrap) that, in most cases, are smaller than the data points.