Quantum Circuits, Inc.

B. Patra, R. M. Incandela, J. V. van Dijk, H. Homulle, Lin Song, M. Shahmohammadi, R. Staszewski, A. Vladimirescu, M. Babaie, F. Sebastiano, E. Charbon

Sumeet Khatri, Ryan Larose, Alexander Poremba, L. Cincio, A. Sornborger, Patrick J. Coles

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 unitaryUand a trainable unitaryVas 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 overlapTr(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.

Carlos Bravo-Prieto, Ryan Larose, M. Cerezo, Yiğit Subaşı, L. Cincio, Patrick J. Coles

Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare |x⟩ such that A|x⟩∝|b⟩. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision ϵ is achieved. Specifically, we prove that C⩾ϵ2/κ2, where C is the VQLS cost function and κ is the condition number of A. We present efficient quantum circuits to estimate C, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of 1024×1024. Finally, we numerically solve non-trivial problems of size up to 250×250. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in ϵ, κ, and the system size N.

Piotr Czarnik, A. Arrasmith, Patrick J. Coles, L. Cincio

Achieving near-term quantum advantage will require accurate estimation of quantum observables despite significant hardware noise. For this purpose, we propose a novel, scalable error-mitigation method that applies to gate-based quantum computers. The method generates training data {Xinoisy,Xiexact} via quantum circuits composed largely of Clifford gates, which can be efficiently simulated classically, where Xinoisy and Xiexact are noisy and noiseless observables respectively. Fitting a linear ansatz to this data then allows for the prediction of noise-free observables for arbitrary circuits. We analyze the performance of our method versus the number of qubits, circuit depth, and number of non-Clifford gates. We obtain an order-of-magnitude error reduction for a ground-state energy problem on 16 qubits in an IBMQ quantum computer and on a 64-qubit noisy simulator.

B. Terhal, D. DiVincenzo

We present evidence that there exist quantum computations that can be carried out in constant depth, using 2-qubit gates, that cannot be simulated classically with high accuracy. We prove that if one can simulate these circuits classically efficiently then the complexity class BQP is contained in AM.