Power of quantum measurement in simulating unphysical operations

TL;DR

This work tackles the simulation of unphysical maps beyond CPTP by replacing classical sampling with measurement-controlled post-processing using a quantum instrument. It proves that the optimal simulation cost equals the diamond norm $\\|\\mathcal E\\|_\\diamond$ for all Hermitian-preserving maps, providing the first universal operational meaning of this norm in this context. A key result is that a single quantum instrument suffices to realize optimal simulations via twisted-channel decompositions, outperforming quasi-probability methods in several tasks such as information recovering and entry-extraction maps. The findings have practical implications for error mitigation and quantum machine learning, and open avenues for applying quantum measurement to broader sampling problems in quantum information processing.

Abstract

The manipulation of quantum states through linear maps beyond quantum operations has many important applications in various areas of quantum information processing. Current methods simulate unphysical maps by sampling physical operations according to classically determined probability distributions. In this work, we show that using quantum measurement instead leads to lower simulation costs for general Hermitian-preserving maps. Remarkably, we establish the equality between the simulation cost and the well-known diamond norm, thus closing a previously known gap and assigning diamond norm a universal operational meaning for all Hermitian-preserving maps. We demonstrate our method in two applications closely related to error mitigation and quantum machine learning, where it exhibits a favorable scaling. These findings highlight the power of quantum measurement in simulating unphysical operations, in which quantum interference is believed to play a vital role. Our work paves the way for more efficient sampling techniques and has the potential to be extended to more quantum information processing scenarios.

Figures (4)

• Figure 1: Difference between QPD and measurement-controlled post-processing. The task is to estimate the expectation value of an observable with respect to a state transformed by a Hermitian-preserving map without knowing the observable nor the state. (a) Estimating the expectation value with QPD. The action on the input state in each round is determined by classical random sampling, which is independent of the input state. (b) Estimating the expectation value with measurement-controlled post-processing. The action on the input state in each round is determined by quantum measurement governed by the Born rule, which takes the input state into consideration.
• Figure 2: Comparison between sampling overheads of QPD and measurement-controlled post-processing for information recovering under common noises at different noise levels. The markers on the solid lines represent the overheads achieved by using twisted channels, i.e., measurement-controlled post-processing, and those on the dashed lines are the overheads achieved by QPD.
• Figure 3: Illustration of entry extraction maps and a comparative evaluation of their implementation using QPD and the twisted channel method. (a) A typical instance of an entry extraction map, which converts an $8 \times 8$ matrix into a $4 \times 4$ matrix that comprises all the extracted entries while preserving their relative locations. (b) A plot comparing the costs of QPD and the twisted channel method across various entry extraction maps. All the maps share a common input dimension and are distinguished by the index sets they extract, as indicated on the horizontal axis.
• Figure 4: Estimation of the expectation value with different numbers of measurement shots. The number of measurement shots is measured in terms of its ratio to $M^* \coloneqq \left\| {\cal D}^\epsilon \right\|_\diamond^2 K(\delta, \varepsilon, O)$, where we set $\epsilon=0.2$, $\delta=0.1$ and $\varepsilon=0.1$. Each blue triangle marks an estimation of the expectation value with the corresponding number of measurement shots, and for each number of measurement shots, we repeat the estimation for $300$ times. The dashed gray line marks the exact expectation value, and the space between the two dashed orange lines is the zone where an estimation is within the set accuracy.

Theorems & Definitions (9)

• Definition 1: Twisted channel
• Theorem 2: One quantum instrument is all you need
• Theorem 3: Diamond norm is the cost
• Proposition 4
• Theorem \ref{theorem:one_quantum_instrument}
• Lemma 6
• Theorem \ref{theorem:diamond_norm}
• Proposition 8
• Definition 9