Quantum Error Mitigated Classical Shadows

Classical shadows enable us to learn many properties of a quantum state $\rho$ with very few measurements. However, near-term and early fault-tolerant quantum computers will only be able to prepare noisy quantum states $\rho$ and it is thus a considerable challenge to efficiently learn properties of an ideal, noise-free state ${\rho }_{\mathrm{id}}$. We consider error mitigation techniques, such as probabilistic error cancelation (PEC), zero noise extrapolation (ZNE), and symmetry verification (SV), which have been developed for mitigating errors in single expected value measurements and generalize them for mitigating errors in classical shadows. We find that PEC is the most natural candidate and thus develop a thorough theoretical framework for PEC shadows with the following rigorous theoretical guarantees: PEC shadows are an unbiased estimator for the ideal quantum state ${\rho }_{\mathrm{id}}$; the sample complexity for simultaneously predicting many linear properties of ${\rho }_{\mathrm{id}}$ is identical to that of the conventional shadows approach up to a multiplicative factor, which is the sample overhead due to error mitigation. Due to efficient postprocessing of shadows, this overhead does not depend directly on the number of qubits but rather grows exponentially with the number of noisy gates. The broad set of tools introduced in this work may be instrumental in exploiting near-term and early fault-tolerant quantum computers: we demonstrate in detailed numerical simulations a range of practical applications of quantum computers that will significantly benefit from our techniques.

Popular Summary

Fundamental principles of quantum mechanics pose limitations on quantum computers and extracting information from them is challenging. For instance, to fully characterize an unknown quantum state, we would require an exponential number of copies of the state. However, practical applications prioritize understanding properties of the state rather than obtaining a complete description.

A relatively new technique, known as classical shadows, allows us to extract many properties of a given quantum state from few measurements. This approach is promising, especially for near-term quantum computers. However, it is anticipated that near-term and early fault-tolerant devices will be limited by noise; using classical shadows on noisy states will only give access to noisy properties.

In this work we solve this issue by combining classical shadows with quantum error-mitigation techniques. Error mitigation involves employing tricks to reduce the error in estimating expectation values at the cost of increasing the number of measurements. We explore different error-mitigation techniques and focus on one that emerges as a natural fit for classical shadows. We show that, even in the presence of realistic noise, it is possible to recover noiseless properties using a noisy quantum computer through our quantum error-mitigated shadows. We support our findings with rigorous mathematical proofs and establish bounds on the number of samples required to achieve a guaranteed precision. We then illustrate how our techniques can be used in practice through a rich set of numerical experiments.

We expect that quantum error-mitigated shadows will be instrumental in practical applications of near-term and early fault-tolerant devices.

Figure 1

In the present work we assume we have only access to a noisy quantum computer (left) such that every circuit we run (left, yellow area) gets corrupted by gate noise (unwanted red-gate elements). We aim to extract properties of a state that would be prepared by an ideal quantum computer (right) with the use of powerful error mitigation techniques. We provide a rigorous theoretical framework for PEC shadows, which effectively allows us to obtain a classical shadow of the ideal quantum state (noise-free shadow) from which we can predict many ideal properties in classical postprocessing (middle blue area, classical computer). In our formalism we run a series of distinct quantum circuit variants (left, yellow area) that cast different classical shadows (noisy shadows) due to gate noise and due to our intentional recovery operations (red-gate elements and their shadows). Under the assumption that the device’s error characteristics have been appropriately learned, we can estimate the noise-free shadow (middle) via classical postprocessing.

Figure 2

Illustration of the light cone of an observable $O$, which is represented by the measurement apparatus and acts only nontrivially on the second (from top) qubit. The orange area indicates the qubits which are contained in the light cone $\mathcal{I}$ of the observable with respect to the ideal quantum circuit (orange boxes) ${\mathcal{U}}_3{\mathcal{U}}_2{\mathcal{U}}_1$. To simplify derivations we assume the gate noise channels ${\mathcal{N}}_k$ are local (light green boxes) such that they are contained within the light cone $\mathcal{I}$ but our results can be extended to nonlocal models via Ref. [54].

Figure 3

(Left) A noisy variational Hamiltonian ansatz is used to prepare the ground state of Eq. (10) but our aim is to learn properties of the noise-free state. (Middle) Energy estimation errors for different noise strengths with conventional shadows (dashed blue, dashed red) and with PEC shadows including readout error mitigation (solid blue, solid red). Bias (gray solid lines) is introduced when the ground-state energy $\mathrm{Tr}(\rho \mathcal{H})$ is directly estimated from the noisy quantum state $\rho$. Error mitigated shadows are unbiased as they estimate $\mathrm{Tr}({\rho }_{\mathrm{id}}\mathcal{H})$. Increasing the circuit error rate $\xi$ (blue versus red) increases the bias in standard shadows (dashed blue versus dashed red) and increases the variance of the error mitigated shadows (solid blue versus solid red). Each data point is an average over ${10}^4$ experiments of a fixed shot budget $N_s$. (Right) Error in simultaneously estimating all three-local Pauli strings without (blue) and with (red) PEC and readout error mitigation—only the 200 observables of the highest estimation error are shown and a circuit error rate $\xi \approx 0.72$ is assumed. Errors are significantly below our rigorous bounds from Theorem 2 (which also take into account the overhead due to readout errors) for PEC shadows but the errors for conventional shadows can be above their respective bounds from Ref. [21] due to bias and readout errors (right end of blue).

Figure 4

Simultaneously estimating all three-local Pauli operators using error extrapolated shadows. We estimated noisy expected values (crosses) from shadows of size $N_s={10}^7$ by increasing the native depolarizing error rate $p={10}^{-3}$ to higher levels $\{2\times {10}^{-3},5\times {10}^{-3}\}$ by randomly sampling noisy circuit variants. Using a linear model function we then extrapolate to zero noise to obtain an error mitigated expectation value close to the ideal ones (disks).

Figure 5

A noisy variational Hamiltonian ansatz is used to prepare the ground state of Eq. (11) whose ideal, noise-free Rényi entropies $R_Q$ we can learn with PEC shadows. We plot purities $\mathrm{Tr}({\rho }_Q^2)$ as a proxy for $R_Q:=-\log \mathrm{Tr}({\rho }_Q^2)$. (Left) Purity heat map in the noiseless case and infinite shot limit. An increasing value indicates that the subsystem $Q$ is less entangled with the remaining qubits. (Middle-right) Absolute error in the purities due to gate noise for a circuit error rate $\xi =0.6$ and due to finite repetition using $N_s={10}^5$. (Middle) Although the entanglement pattern is approximately recovered with conventional shadows, in some instances we observe substantial errors, i.e., the largest error is $0.27$. (Right) Absolute errors with PEC shadows are significantly smaller, i.e., the largest error is $7\times {10}^{-2}$ but this figure could be further reduced by increasing $N_s$.