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We propose a novel deterministic method for preparing arbitrary quantum states. When our protocol is compiled into CNOT and arbitrary single-qubit gates, it prepares an$N$-dimensional state in depth$O(\log (N))$and$\mathit{\text{spacetime allocation}}$(a metric that accounts for the fact that oftentimes some ancilla qubits need not be active for the entire circuit)$O(N)$, which are both optimal. When compiled into the$\{\mathrm{H},\mathrm{S},\mathrm{T},\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T}\}$gate set, we show that it requires asymptotically fewer quantum resources than previous methods. Specifically, it prepares an arbitrary state up to error$ϵ$with optimal depth of$O(\log (N)+\log (1/ϵ))$and spacetime allocation$O(N\log (\log (N)/ϵ))$, improving over$O(\log (N)\log (\log (N)/ϵ))$and$O(N\log (N/ϵ))$, respectively. We illustrate how the reduced spacetime allocation of our protocol enables rapid preparation of many disjoint states with only constant-factor ancilla overhead –$O(N)$ancilla qubits are reused efficiently to prepare a product state of$w$$N$-dimensional states in depth$O(w+\log (N))$rather than$O(w\log (N))$, achieving effectively constant depth per state. We highlight several applications where this ability would be useful, including quantum machine learning, Hamiltonian simulation, and solving linear systems of equations. We provide quantum circuit descriptions of our protocol, detailed pseudocode, and gate-level implementation examples using Braket.

 

Simulating the time evolution of a physical system at quantum mechanical levels of detail - known as Hamiltonian Simulation (HS) - is an important and interesting problem across physics and chemistry. For this task, algorithms that run on quantum computers are known to be exponentially faster than classical algorithms; in fact, this application motivated Feynman to propose the construction of quantum computers. Nonetheless, there are challenges in reaching this performance potential. Prior work has focused on compiling circuits (quantum programs) for HS with the goal of maximizing either accuracy or gate cancellation. Our work proposes a compilation strategy that simultaneously advances both goals. At a high level, we use classical optimizations such as graph coloring and travelling salesperson to order the execution of quantum programs. Specifically, we group together mutually commuting terms in the Hamiltonian (a matrix characterizing the quantum mechanical system) to improve the accuracy of the simulation. We then rearrange the terms within each group to maximize gate cancellation in the final quantum circuit. These optimizations work together to improve HS performance and result in an average 40% reduction in circuit depth. This work advances the frontier of HS which in turn can advance physical and chemical modeling in both basic and applied sciences. 

Variational quantum eigensolver (VQE) is a promising algorithm suitable for near-term quantum computers. VQE aims to approximate solutions to exponentially-sized optimization problems by executing a polynomial number of quantum subproblems. However, the number of subproblems scales as N 4 for typical problems of interest-a daunting growth rate that poses a serious limitation for emerging applications such as quantum computational chemistry. We mitigate this issue by exploiting the simultaneous measurability of subproblems corresponding to commuting terms. Our technique transpiles VQE instances into a format optimized for simultaneous measurement, ultimately yielding 8-30x lower cost. Our work also encompasses a synthesis tool for compiling simultaneous measurement circuits with minimal overhead. We demonstrate experimental validation of our techniques by estimating the ground state energy of deuteron with a quantum computer. We also investigate the underlying statistics of simultaneous measurement and devise an adaptive strategy for mitigating harmful covariance terms.