Quantum-assisted quantum compiling

Sumeet Khatri1,2, Ryan LaRose1,3, Alexander Poremba1,4, Lukasz Cincio1, Andrew T. Sornborger5, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM USA.
2Hearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA USA.
3Department of Computational Mathematics, Science, and Engineering and Department of Physics and Astronomy, Michigan State University, East Lansing, MI USA.
4Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA USA.
5Information Sciences, Los Alamos National Laboratory, Los Alamos, NM USA.

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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 ${\rm Tr}(V^†U)$ 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.

Ordinary computers require a compiler that converts one's code into a machine-level language. Quantum computers require a compiler as well. However, a new challenge for such "quantum compilers" is that they should be optimal, i.e., they should return a machine-level program that has as few operations as possible. This optimality is crucial for current noisy quantum devices, where longer programs accumulate more errors while shorter programs avoid errors. In this work, we introduce an algorithm for optimal quantum compiling. The key feature that allows for optimality is that we propose to use quantum computers themselves to assist in the compiling process. Hence, our algorithm is called quantum-assisted quantum compiling (QAQC, pronounced "Quack").

The idea is that one needs to quantify the distance between the original program and the compiled program, with the goal of trying to minimize this distance. We prove that this distance calculation cannot be done efficiently on a classical computer. On the other hand, we provide an efficient quantum circuit for computing it.

In addition to shortening the length of one's quantum program, QAQC can be used to learn algorithms that compensate for a given quantum computer's noise and also to benchmark the noise processes occurring on a quantum computer. We successfully implement QAQC for small programs using currently available quantum computers from IBM and Rigetti, and we use simulators to explore the compilation of larger programs. Overall, QAQC appears to be a promising tool for mitigating errors in the era of noisy intermediate-scale quantum computers.

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