[Paper Review] How to use arbitrary measuring devices to perform almost perfect measurements

Noah Linden, Paul Skrzypczyk|arXiv (Cornell University)|Mar 4, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a general framework for using imperfect or arbitrary quantum measuring devices multiple times to approximate a target quantum measurement with exponentially decreasing error. By leveraging classical cloning, block coding, and probabilistic protocols, it shows that any generalized quantum measurement can be asymptotically reproduced with perfect fidelity, revealing that all non-trivial measurements are asymptotically equivalent under multiple uses.

ABSTRACT

We consider the problem of reproducing one quantum measurement given the ability to perform another. We give a general framework and specific protocols for this problem. For example, we show how to use available "imperfect" devices a small number of times to implement a target measurement with average error that drops off exponentially with the number of imperfect measurements used. We hope that could be useful in near-term applications as a type of lightweight error mitigation of the measuring devices. As well as the view to practical applications, we consider the question from a general theoretical perspective in the most general setting where both the available and target measurements are arbitrary generalised quantum measurements. We show that this general problem in fact reduces to the ability to reproduce the statistics of (complete) von Neumann measurements, and that in the asymptotic limit of infinitely many uses of the available measurement, a simple protocol based upon `classical cloning' can perfectly achieve this task. We show that asymptotically all (non-trivial) quantum measurements are equivalent. We also study optimal protocols for a fixed number of uses of the available measurement. This includes, but is not limited to, improving both noisy and lossy quantum measurements. Furthermore, we show that, in a setting where we perform multiple measurements in parallel, we can achieve finite-rate measurement reproduction, by using block-coding techniques from classical information theory. Finally, we show that advantages can also be gained by making use of probabilistic protocols.

Motivation & Objective

• To develop a general protocol for approximating a target quantum measurement using an arbitrary, potentially imperfect or noisy measuring device.
• To minimize the average error in measurement approximation by leveraging multiple uses of the available device.
• To explore the theoretical limits of measurement interconversion under multiple-use protocols, especially in the asymptotic regime.
• To investigate practical applications in near-term quantum technologies through lightweight error mitigation.
• To extend resource-theoretic frameworks by allowing multiple uses of a measurement resource, thereby enabling finite-rate measurement reproduction.

Proposed method

• Using classical cloning to distribute quantum state information across multiple copies before measuring each with the available device.
• Applying block-coding techniques from classical information theory to achieve finite-rate measurement reproduction over multiple uses.
• Designing probabilistic protocols that can outperform deterministic ones in certain scenarios, particularly for extremal or non-Gaussian measurements.
• Reducing the general problem of measurement interconversion to the task of reproducing von Neumann measurement statistics.
• Employing unitary operations and ancillary systems to coherently process the input state prior to measurement, enabling flexible manipulation of measurement outcomes.
• Using average root-mean-squared error as the primary figure of merit to optimize protocols across different measurement types.

Experimental results

Research questions

• RQ1 Can arbitrary imperfect measuring devices be used multiple times to implement a target measurement with exponentially decreasing error?
• RQ2 What is the optimal protocol for approximating a target measurement when only a fixed number of uses of the available device are allowed?
• RQ3 Can finite-rate measurement reproduction be achieved using block-coding schemes inspired by classical Shannon theory?
• RQ4 How does the performance of classical-cloning-based protocols compare to more sophisticated protocols for extremal or non-trivial measurements?
• RQ5 To what extent can measurement interconversion be achieved when only partial information about the available device is known?

Key findings

• The average error in approximating a target measurement decreases exponentially with the number of uses of the available device.
• All non-trivial generalized quantum measurements become asymptotically equivalent under multiple uses, with perfect reproduction achievable in the limit of infinite uses.
• Classical-cloning-based protocols are not always optimal, as demonstrated by superior performance in the trine measurement case.
• Finite-rate measurement reproduction is possible using block-coding techniques derived from the noisy-channel coding theorem.
• Probabilistic protocols can outperform deterministic ones, especially when dealing with extremal or highly non-Gaussian measurements.
• The framework generalizes resource theories of measurements by allowing multiple uses of a resourceful measurement, thereby enriching the structure of measurement interconversions.