The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer enables the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, and phase estimation. The standard fault-tolerant implementation of an n-qubit QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+t gates, incurring the t-count complexity of O(n log2 (n)). In this paper we show how to obtain approximate QFT with the t-count of O(n log(n)). Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.

1 aNam, Yunseong1 aSu, Yuan1 aMaslov, Dmitri uhttps://arxiv.org/abs/1803.0493301650nas a2200157 4500008004100000245006100041210006100102300001400163520116300177100002301340700001901363700001801382700001901400700001301419856006001432 2018 eng d00aToward the first quantum simulation with quantum speedup0 aToward the first quantum simulation with quantum speedup a2018017233 aWith quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, using diverse techniques to tighten error bounds and optimize circuit implementations. Quantum signal processing appears to be preferred among algorithms with rigorous performance guarantees, whereas higher-order product formulas prevail if empirical error estimates suffice. Our circuits are orders of magnitude smaller than those for the simplest classically infeasible instances of factoring and quantum chemistry, bringing practical quantum computation closer to reality.

1 aChilds, Andrew, M.1 aMaslov, Dmitri1 aNam, Yunseong1 aRoss, Neil, J.1 aSu, Yuan uhttp://www.pnas.org/content/early/2018/09/04/180172311501277nas a2200157 4500008004100000245008000041210006900121260001500190520078500205100001800990700001901008700001301027700002301040700001901063856003701082 2017 eng d00aAutomated optimization of large quantum circuits with continuous parameters0 aAutomated optimization of large quantum circuits with continuous c2017/10/193 aWe develop and implement automated methods for optimizing quantum circuits of the size and type expected in quantum computations that outperform classical computers. We show how to handle continuous gate parameters and report a collection of fast algorithms capable of optimizing large-scale quantum circuits. For the suite of benchmarks considered, we obtain substantial reductions in gate counts. In particular, we provide better optimization in significantly less time than previous approaches, while making minimal structural changes so as to preserve the basic layout of the underlying quantum algorithms. Our results help bridge the gap between the computations that can be run on existing hardware and those that are expected to outperform classical computers.

1 aNam, Yunseong1 aRoss, Neil, J.1 aSu, Yuan1 aChilds, Andrew, M.1 aMaslov, Dmitri uhttps://arxiv.org/abs/1710.0734501713nas a2200133 4500008004100000245007300041210006900114260001500183300001100198490000700209520120600216100001901422856013801441 2017 eng d00aBasic circuit compilation techniques for an ion-trap quantum machine0 aBasic circuit compilation techniques for an iontrap quantum mach c2016/02/20 a0230350 v193 aWe study the problem of compilation of quantum algorithms into optimized physical-level circuits executable in a quantum information processing (QIP) experiment based on trapped atomic ions. We report a complete strategy: starting with an algorithm in the form of a quantum computer program, we compile it into a high-level logical circuit that goes through multiple stages of decomposition into progressively lower-level circuits until we reach the physical execution-level specification. We skip the fault-tolerance layer, as it is not necessary in this work. The different stages are structured so as to best assist with the overall optimization while taking into account numerous optimization criteria, including minimizing the number of expensive two-qubit gates, minimizing the number of less expensive single-qubit gates, optimizing the runtime, minimizing the overall circuit error, and optimizing classical control sequences. Our approach allows a trade-off between circuit runtime and quantum error, as well as to accommodate future changes in the optimization criteria that may likely arise as a result of the anticipated improvements in the physical-level control of the experiment.

1 aMaslov, Dmitri uhttp://iopscience.iop.org/article/10.1088/1367-2630/aa5e47/meta;jsessionid=55CC235A0B106081E825099310586F07.c3.iopscience.cld.iop.org02065nas a2200169 4500008004100000245007000041210006900111260001500180520154900195100001601744700001901760700002101779700001801800700001601818700002401834856003701858 2017 eng d00aComplete 3-Qubit Grover Search on a Programmable Quantum Computer0 aComplete 3Qubit Grover Search on a Programmable Quantum Computer c2017/03/303 aSearching large databases is an important problem with broad applications. The Grover search algorithm provides a powerful method for quantum computers to perform searches with a quadratic speedup in the number of required database queries over classical computers. It is an optimal search algorithm for a quantum computer, and has further applications as a subroutine for other quantum algorithms. Searches with two qubits have been demonstrated on a variety of platforms and proposed for others, but larger search spaces have only been demonstrated on a non-scalable NMR system. Here, we report results for a complete three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. The algorithm is performed for all 8 possible single-result oracles and all 28 possible two-result oracles. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state-marking scheme required to perform a classical search. All quantum solutions are shown to outperform their classical counterparts. We also report the first implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5% and 89.6%, respectively.

1 aFiggatt, C.1 aMaslov, Dmitri1 aLandsman, K., A.1 aLinke, N., M.1 aDebnath, S.1 aMonroe, Christopher uhttps://arxiv.org/abs/1703.1053501709nas a2200229 4500008004100000245006700041210006700108250000700175260001500182300001400197490000800211520106700219100001601286700001901302700002201321700001601343700001601359700002101375700001501396700002401411856004401435 2017 eng d00aExperimental Comparison of Two Quantum Computing Architectures0 aExperimental Comparison of Two Quantum Computing Architectures a13 c2017/03/21 a3305-33100 v1143 aWe run a selection of algorithms on two state-of-the-art 5-qubit quantum computers that are based on different technology platforms. One is a publicly accessible superconducting transmon device [1] with limited connectivity, and the other is a fully connected trapped-ion system [2]. Even though the two systems have different native quantum interactions, both can be programmed in a way that is blind to the underlying hardware, thus allowing the first comparison of identical quantum algorithms between different physical systems. We show that quantum algorithms and circuits that employ more connectivity clearly benefit from a better connected system of qubits. While the quantum systems here are not yet large enough to eclipse classical computers, this experiment exposes critical factors of scaling quantum computers, such as qubit connectivity and gate expressivity. In addition, the results suggest that co-designing particular quantum applications with the hardware itself will be paramount in successfully using quantum computers in the future.

1 aLinke, N.M.1 aMaslov, Dmitri1 aRoetteler, Martin1 aDebnath, S.1 aFiggatt, C.1 aLandsman, K., A.1 aWright, K.1 aMonroe, Christopher uhttp://www.pnas.org/content/114/13/330501977nas a2200121 4500008004100000245009300041210006900134260001500203520155900218100001901777700002201796856003701818 2017 eng d00aShorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations0 aShorter stabilizer circuits via Bruhat decomposition and quantum c2017/05/253 aIn this paper we improve the layered implementation of arbitrary stabilizer circuits introduced by Aaronson and Gottesman in Phys. Rev. A 70(052328), 2004: to implement a general stabilizer circuit, we reduce their 11-stage computation -HC-P-C-P-C-H-P-C-P-C- over the gate set consisting of Hadamard, Controlled-NOT, and Phase gates, into a 7-stage computation of the form -C-CZ-P-H-P-CZ-C-. We show arguments in support of using -CZ- stages over the -C- stages: not only the use of -CZ- stages allows a shorter layered expression, but -CZ- stages are simpler and appear to be easier to implement compared to the -C- stages. Based on this decomposition, we develop a twoqubit gate depth-(14n−4) implementation of stabilizer circuits over the gate library {H, P, CNOT}, executable in the LNN architecture, improving best previously known depth-25n circuit, also executable in the LNN architecture. Our constructions rely on Bruhat decomposition of the symplectic group and on folding arbitrarily long sequences of the form (-P-C-) m into a 3-stage computation -P-CZ-C-. Our results include the reduction of the 11-stage decomposition -H-C-P-C-P-C-H-P-C-P-C- into a 9-stage decomposition of the form -C-P-C-P-H-C-P-C-P-. This reduction is based on the Bruhat decomposition of the symplectic group. This result also implies a new normal form for stabilizer circuits. We show that a circuit in this normal form is optimal in the number of Hadamard gates used. We also show that the normal form has an asymptotically optimal number of parameters.

1 aMaslov, Dmitri1 aRoetteler, Martin uhttps://arxiv.org/abs/1705.0917601522nas a2200157 4500008004100000245006100041210006100102260001500163520105700178100002301235700001901258700001801277700001901295700001301314856003701327 2017 eng d00aToward the first quantum simulation with quantum speedup0 aToward the first quantum simulation with quantum speedup c2017/11/293 aWith quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, employing diverse techniques to tighten error bounds and optimize circuit implementations. Quantum signal processing appears to be preferred among algorithms with rigorous performance guarantees, whereas higher-order product formulas prevail if empirical error estimates suffice. Our circuits are orders of magnitude smaller than those for the simplest classically-infeasible instances of factoring and quantum chemistry.

1 aChilds, Andrew, M.1 aMaslov, Dmitri1 aNam, Yunseong1 aRoss, Neil, J.1 aSu, Yuan uhttps://arxiv.org/abs/1711.1098000962nas a2200121 4500008004100000245007400041210006900115260001500184520054100199100001900740700001800759856006300777 2017 eng d00aUse of global interactions in efficient quantum circuit constructions0 aUse of global interactions in efficient quantum circuit construc c2017/12/213 aIn this paper we study the ways to use a global entangling operator to efficiently implement circuitry common to a selection of important quantum algorithms. In particular, we focus on the circuits composed with global Ising entangling gates and arbitrary addressable single-qubit gates. We show that under certain circumstances the use of global operations can substantially improve the entangling gate count.

1 aMaslov, Dmitri1 aNam, Yunseong uhttp://iopscience.iop.org/article/10.1088/1367-2630/aaa39801659nas a2200133 4500008004100000245011600041210006900157260001500226300001100241490000700252520121100259100001901470856003601489 2016 eng d00aOn the advantages of using relative phase Toffolis with an application to multiple control Toffoli optimization0 aadvantages of using relative phase Toffolis with an application c2016/02/10 a0223110 v933 aVarious implementations of the Toffoli gate up to a relative phase have been known for years. The advantage over regular Toffoli gate is their smaller circuit size. However, their use has been often limited to a demonstration of quantum control in designs such as those where the Toffoli gate is being applied last or otherwise for some specific reasons the relative phase does not matter. It was commonly believed that the relative phase deviations would prevent the relative phase Toffolis from being very helpful in practical large-scale designs. In this paper, we report three circuit identities that provide the means for replacing certain configurations of the multiple control Toffoli gates with their simpler relative phase implementations, up to a selectable unitary on certain qubits, and without changing the overall functionality. We illustrate the advantage via applying those identities to the optimization of the known circuits implementing multiple control Toffoli gates, and report the reductions in the CNOT-count, T-count, as well as the number of ancillae used. We suggest that a further study of the relative phase Toffoli implementations and their use may yield other optimizations.1 aMaslov, Dmitri uhttp://arxiv.org/abs/1508.0327300852nas a2200133 4500008004100000245008100041210006900122260001500191300001400206490000700220520043600227100001900663856003600682 2016 eng d00aOptimal and asymptotically optimal NCT reversible circuits by the gate types0 aOptimal and asymptotically optimal NCT reversible circuits by th c2016/08/23 a1096-11120 v163 aWe report optimal and asymptotically optimal reversible circuits composed of NOT, CNOT, and Toffoli (NCT) gates, keeping the count by the subsets of the gate types used. This study fine tunes the circuit complexity figures for the realization of reversible functions via reversible NCT circuits. An important consequence is a result on the limitation of the use of the T-count quantum circuit metric popular in applications.

1 aMaslov, Dmitri uhttp://arxiv.org/abs/1602.0262701464nas a2200169 4500008004100000022001400041245010200055210006900157260001500226300001400241490000700255520082200262100002301084700001901107700001901126856014901145 2016 eng d a0018-934000aPractical Approximation of Single-Qubit Unitaries by Single-Qubit Quantum Clifford and T Circuits0 aPractical Approximation of SingleQubit Unitaries by SingleQubit c2016/01/01 a161 - 1720 v653 aWe present an algorithm, along with its implementation that finds T-optimal approximations of single-qubit Z-rotations using quantum circuits consisting of Clifford and T gates. Our algorithm is capable of handling errors in approximation down to size 10-15, resulting in the optimal single-qubit circuit designs required for implementation of scalable quantum algorithms. Our implementation along with the experimental results are available in the public domain.

1 aKliuchnikov, Vadym1 aMaslov, Dmitri1 aMosca, Michele uhttp://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7056491http://xplorestaging.ieee.org/ielx7/12/7350319/7056491.pdf?arnumber=7056491