Quantum-assisted quantum compiling

Abstract

Compiling quantum algorithms for near-term quantum computers (accounting for connectivity and native gate alphabets) is a major challenge that has received significant attention both by industry and academia. Avoiding the exponential overhead of classical simulation of quantum dynamics will allow compilation of larger algorithms, and a strategy for this is to evaluate an algorithm's cost on a quantum computer. To this end, we propose a variational hybrid quantum-classical algorithm called quantum-assisted quantum compiling (QAQC). In QAQC, we use the overlap between a target unitary U and a trainable unitary V as the cost function to be evaluated on the quantum computer. More precisely, to ensure that QAQC scales well with problem size, our cost involves not only the global overlap Tr(V † U ) but also the local overlaps with respect to individual qubits. We introduce novel short-depth quantum circuits to quantify the terms in our cost function, and we prove that our cost cannot be efficiently approximated with a classical algorithm under reasonable complexity assumptions. We present both gradient-free and gradient-based approaches to minimizing this cost. As a demonstration of QAQC, we compile various one-qubit gates on IBM's and Rigetti's quantum computers into their respective native gate alphabets. Furthermore, we successfully simulate QAQC up to a problem size of 9 qubits, and these simulations highlight both the scalability of our cost function as well as the noise resilience of QAQC. Future applications of QAQC include algorithm depth compression, black-box compiling, noise mitigation, and benchmarking. arXiv:1807.00800v5 [quant-ph]

Noise resilience of variational quantum compiling

“…On the other hand, F j LHST ( ) is the entanglement fidelity for the quantum channel obtained from feeding into V U † the maximally mixed state on A j and then tracing over A j , where A j consists of all qubits in A other than A j . As shown in [19] …”

“…One is to compile the full unitary matrix U by considering the action of U on all input states (or an informationally complete set of states) [19,21]. The other is to compile only a particular column of the matrix U by considering the action of U on a fixed input state [19,20]. The benefit of the first approach is that it is fully general, applying even when one does not know what the input state to U will be (for example, if U occurs in the middle of one's quantum algorithm).…”

“…A slightly different but equivalent approach was employed in [21]. We focus on the approach of [19] in what follows.…”

“…Two cost functions were considered in [19]. One cost function C HST quantifies the Hilbert-Schmidt inner product between the target unitary U and the trainable gate sequence V, as follows:…”

“…Note that C HST is faithful in that = C 0 HST iff V=U (up to a global phase). An alternative cost function [19] is given by…”