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arXiv:2109.09629

A short note on the 0-fidelity

Karl Mayer

Published Sep 20, 2021• 0 citations
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Abstract

A recent article introduced a hierarchy of quantities called $k$-fidelities that approximate the quantum process fidelity with increasing accuracy. The lowest approximation in this hiearchy is the $0$-fidelity. The authors gave a protocol for estimating the $0$-fidelity and showed numerical evidence that it approximates the process fidelity. In this note, we prove lower and upper bounds on the process fidelity as linear functions of the 0-fidelity. By solving a semidefinite program, we provide evidence that the lower bound is tight.

DOI
10.48550/arxiv.2109.09629

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A recent article introduced a hierarchy of quantities called kkk-fidelities that approximate the quantum process fidelity with increasing accuracy. The lowest approximation in this hiearchy is the 000-fidelity. The authors gave a protocol for estimating the 000-fidelity and showed numerical evidence that it approximates the process fidelity. In this note, we prove lower and upper bounds on the process fidelity as linear functions of the 0-fidelity. By solving a semidefinite program, we provide evidence that the lower bound is tight.
The accurate implementation of quantum gates is essential for the realization of quantum algorithms and digital quantum simulations. This accuracy may be increased on noisy hardware through the variational optimization of gates, however the experimental realization of such a protocol is impeded by the large effort required to estimate the fidelity of an implemented gate. With a hierarchy of approximations we find a faithful approximation to the quantum process fidelity that can be estimated experimentally with reduced effort. Its practical use is demonstrated with the optimization of a three-qubit quantum gate on a commercially available quantum processor.
We investigate the problem of bounding the quantum process fidelity given bounds on the fidelities between target states and the action of a process on a set of pure input states. We formulate the problem as a semidefinite program and prove convexity of the minimum process fidelity as a function of the errors on the output states. We characterize the conditions required to uniquely determine a process in the case of no errors and derive a lower bound on its fidelity in the limit of small errors for any set of input states satisfying these conditions. We then consider sets of input states whose one-dimensional projectors form a symmetric positive operator-valued measure (POVM). We prove that for such sets the minimum fidelity is bounded by a linear function of the average output state error. A symmetric POVM with minimal number of elements contains d+1d+1d+1 states, where ddd is the Hilbert space dimension. Our bounds applied to such states provide an efficient method for estimating the process fidelity without the use of full process tomography.
In recent years, semidefinite programming has played a vital role in shaping complexity theory and quantum computing. There have been numerous applications ranging from estimating quantum values, over approximating combinatorial quantities, to proving various bounds. This work extends the use of semidefinite programs (SDPs) to proving product rules and to characterizing quantum query complexity. In the first application, we provide a general framework to establishing product rules for quantities that can be expressed (or approximated) using SDPs. We use duality theory to give product rules, which bound the value of the ``product'' of two problems in terms of their value. Some previous results have implicitly used the properties of SDPs to give such product rules. Here we give sufficient and necessary conditions under which these approaches work, thereby enabling us to capture these previous results under our unified framework. We also include a discussion about alternate definitions of what a ``product'' means and how they fit into our approach. The second application provides an SDP characterization of quantum query complexity, which is one of the ways in which complexity of a function can be measured. It is known that quantum query complexity can be lower bounded by the so-called ``adversary method'' which is expressible as a semidefinite program. Recently, Ben Reichardt showed that the adversary method leads to a tight lower bound for boolean functions by converting the solution of this SDP (of adversary method) into an algorithm. We show that a related SDP, called ``witness size'' in this thesis, provides a tight bound on the quantum query complexity of non boolean functions (total as well as partial). This witness size SDP is also used to give composition results for quantum query complexity. We also show that the witness size is bounded by a constant multiple of the adversary bound. Finally, we briefly explore whether other convex programming paradigms can be useful in complexity theory. One of them is copositive programming. We show that one of the recent result about parallel repetition of unique games, by Barak et.al., can be interpreted as an application of copositive programming.
Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of εεε in poly(1/ε,input dimension)\mathrm{poly}(1/ε, \text{input dimension})poly(1/ε,input dimension) time.