On the controlled-NOT complexity of controlled-NOT–phase circuits

Abstract

We study the problem of CNOT-optimal quantum circuit synthesis over gate sets consisting of CNOT and Z-basis rotations of arbitrary angles. We show that the circuit-polynomial correspondence relates such circuits to Fourier expansions of pseudo-Boolean functions, and that for certain classes of functions this expansion uniquely determines the minimum CNOT cost of an implementation. As a corollary we prove that CNOT minimization over CNOT and phase gates is at least as hard as synthesizing a CNOT-optimal circuit computing a set of parities of its inputs. We then show that this problem is NP-complete for two restricted cases where all CNOT gates are required to have the same target, and where the circuit inputs are encoded in a larger state space. The latter case has applications to CNOT optimization over more general Clifford+T circuits. We further present an efficient heuristic algorithm for synthesizing circuits over CNOT and Z-basis rotations with small CNOT cost. Our experiments show a 23% reduction of CNOT gates on average across a suite of Clifford+T benchmark circuits, with a maximum reduction of 43%.

Towards Large-scale Functional Verification of Universal Quantum Circuits

“…We used our algorithm to verify a suite of optimized benchmark circuits against their original input. For the optimization algorithm we chose the GRAYSYNTH algorithm from [1] which is implemented in FEYNMAN and verified each benchmark reported in that paper. Table 1 reports the results of our experiments.…”

“…The phase polynomial could instead be represented in linear space for any k by its Fourier expansion [1,26]. This however complicates the process of verification as the Fourier expansion is not necessarily unique modulo integer multiples [1]. A possible compromise would be to store the Fourier expansion normally, and generate the multilinear form for small subsets on demand.…”

“…The reasoning above applies to any situation where an internal path variable y i only appears with coefficients taken from the Boolean subgroup {0, 1 2 } of D/Z, as the two branches of y i are identical, except that y i = 1 path picks up a multiplicative factor of −1 whenever the quotient of P/y i is odd. Specifically, it can be shown that…”

“…In this work we propose a novel framework for the formal specification and functional verification of unitary (i.e., measurement-free) quantum circuits over a universal gate set -specifically, the Clifford group extended with Z-axis rotations taken from the Clifford hierarchy [17]. Our framework is built around Richard Feynman's path integral technique, which has been used recently to prove results in complexity theory [12,24], and to perform circuit simulation [10,21] and optimization [1,2,4]. Specifically, we develop a concrete representation of quantum operators as path-sums -exponential sums of basis states over a finite set of Boolean path variables.…”

Software Mitigation of Crosstalk on Noisy Intermediate-Scale Quantum Computers

“…Nodes are physical qubits, and edges are possible CNOT gates. When a CNOT is executed on qubits (0, 1) and another CNOT is executed simultaneously on qubits (2,3), the error rate of both CNOTs increases because of crosstalk. Qubit 2 has low coherence, which means that long computation on that qubit (including any idle time after the first operation) is highly error prone.…”