Provable bounds for noise-free expectation values computed from noisy samples

TL;DR

The sampling overhead to extract good samples from noisy quantum computers is formally quantify to extract good samples from noisy quantum computers and relate it to the layer fidelity, a metric to determine the performance of noisy quantum processors.

Abstract

In this paper, we explore the impact of noise on quantum computing, particularly focusing on the challenges when sampling bit strings from noisy quantum computers as well as the implications for optimization and machine learning applications. We formally quantify the sampling overhead to extract good samples from noisy quantum computers and relate it to the layer fidelity, a metric to determine the performance of noisy quantum processors. Further, we show how this allows us to use the Conditional Value at Risk of noisy samples to determine provable bounds on noise-free expectation values. We discuss how to leverage these bounds for different algorithms and demonstrate our findings through experiments on a real quantum computer involving up to 127 qubits. The results show a strong alignment with theoretical predictions.

Figures (6)

• Figure 1: QAOA results on 40-qubits. The curve is the cumulative distributions function resulting from sampling the circuits for a MAXCUT instance executed on ibm_sherbrooke for $p=1$ with $10^5$ shots (top) and $p=2$ with $10^7$ shots (bottom). The vertical lines show the corresponding noisy expectation values (dashed blue), the noise-free expectation values evaluated using light-cone optimized classical simulation (cyan dashed-dotted), the $\overline{\mathop{\mathrm{CVaR}}\nolimits}_{\alpha_p}$ (cyan dotted), and the globally optimal solution equal to $56$ (green solid). The title shows the fitted $\alpha_p'$ such that the $\overline{\mathop{\mathrm{CVaR}}\nolimits}_{\alpha_p'}$ are equal to the noise-free expectation values (i.e. cyan dashed-dotted).
• Figure 2: Example heavy-hex hardware compatible $127$ qubit higher order Ising model. Nodes denote the linear terms, edges between nodes denote the quadratic terms, and the ovals encircling three neighboring nodes on the hardware graph denote hyper-edges. Polynomial (Ising model) coefficients of $-1$ are denoted by red, and $+1$ are denoted by blue.
• Figure 3: QAOA results for sampling a random hardware-compatible higher-order Ising model (minimization combinatorial optimization problem) on 127-qubits: This figure shows the resulting distributions from 127-qubit circuits executed on ibm_sherbrooke for $p=1,\ldots,5$ (top to bottom). The cumulative distribution functions show the values of the resulting samples from $10^5$ shots for every $p$. The vertical lines show the corresponding noisy expectation values (dashed blue), the noise-free expectation values evaluated using MPS simulation (cyan dashed-dotted), and the globally optimal solution equal to $-188$ (green solid). The title shows the fitted $\alpha_p'$ such that the $\mathop{\mathrm{CVaR}}\nolimits_{\alpha_p'}$ are equal to the noise-free expectation values (i.e. cyan dashed-dotted). The corresponding $\alpha_p'$ are indicated by the horizontal dashed red line.
• Figure 4: Variance of CVaR estimates: We draw $10^5$ uniform samples from the original data to estimate the CVaR for $\alpha_p'$, $p=1, \ldots 5$, cf. Tab. \ref{['tab:127_qubit_results']}, and repeat this $10^4$ times to get an estimate of the variance of the CVaR estimator. The dashed green line is fitted to the results for $p=3, \ldots, 5$, and is very close to the predicted behavior of $\mathcal{O}(1/\alpha)$.
• Figure 5: QAOA results on 127 qubits run on ibm_sherbrooke. This figure shows the cumulative distribution functions (CDFs) for depths $p=1,2$ with and without twirling. The orange lines correspond to the twirled circuits, and the blue lines correspond to the untwirled circuits. For each method, $100,000$ shots were used in total. When twirling, we sampled $1,000$ random twirls and performed $100$ shots for each. The statistics of these distributions are summarized in Table \ref{['table:obj-vals-127']}.

Theorems & Definitions (6)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof