Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors

Abstract

Mitigating errors in quantum information processing devices is especially important in the absence of fault tolerance. An effective method in suppressing state-preparation errors is using multiple copies to distill the ideal component from a noisy quantum state. Here, we use classical shadows and randomized measurements to circumvent the need for coherent access to multiple copies at an exponential cost. We study the scaling of resources using numerical simulations and find that the overhead is still favorable compared to full state tomography. We optimize measurement resources under realistic experimental constraints and apply our method to an experiment preparing Greenberger-Horne-Zeilinger (GHZ) state with trapped ions. In addition to improving stabilizer measurements, the analysis of the improved results reveals the nature of errors affecting the experiment. Hence, our results provide a directly applicable method for mitigating errors in near-term quantum computers.

Improving the efficiency of learning-based error mitigation

“…The additional gates are introduced probabilistically according to the device noise model in order to cancel noise effects. An alternative approach is to use multiple copies of a noisy state to prepare its purification [17][18][19][20][21][22][23][24]. Such an approach is called Virtual Distillation.…”

Mitigating algorithmic errors in quantum optimization through energy extrapolation

“…A technique called classical shadows, which has been used for hardware EM [51], can reduce this cost efficiently [52,53]. In our protocol, if we construct shadows via random Pauli measurements and use them to predict the local expectation values simultaneously, the number of measurements required is of order O(3 k log(m)/ 2 ) to estimate the local energy terms Tr(ρ f h[j]) up to additive error , and is of order O(3 2k log(m 2 )/ 2 ) to estimate the local variance terms Tr(ρ f h[i]h[j]) up to .…”

Virtual distillation with noise dilution

“…Recipes to implement such a measurement from multiple copies of ρ ′ using two-qubit entangling gates have been proposed [67,68,[87][88][89][90][91][92]. More recently, a separate idea of using shadow tomography [70,[93][94][95][96] as an efficient post-processing protocol for estimating tr ρ ′M O /tr ρ ′M with only randomized single-qubit unitary rotations in addition to the VQA circuit further enhances the feasibility of this scheme. The name virtual distillation is fitting, since all practical implementations never directly generate the distilled state ρ ′M /tr ρ ′M , but only estimate the corresponding observable measurements.…”

“…Virtual distillation [67][68][69] is yet another technique that can mitigate noise of small error rates in a model-agnostic manner. Moreover, mitigation happens on-the-fly either with an external correcting circuit [68] or with efficient data post-processing using shadow tomography [70] that requires only randomized single-qubit unitary gates. Consequently, this technique can cope with noise drifts, which is an attractive feature in addition to its technical simplicity that permits accessible analysis.…”