Matthew Amy

A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits

We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speedup over simple brute force algorithms. As an illustration of our method, we implemented this algorithm and found factorizations of commonly used quantum logical operations into elementary gates in the Clifford+T set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and T gates. Our decomposition achieves a total T-depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate. Due to the size of the search space, the algorithm is only practical for small parameters, such as the number of qubits, and the number of gates in an optimal implementation.

Polynomial-Time T-Depth Optimization of Clifford+T Circuits Via Matroid Partitioning

Most work in quantum circuit optimization has been performed in isolation from the results of quantum fault-tolerance. Here we present a polynomial-time algorithm for optimizing quantum circuits that takes the actual implementation of fault-tolerant logical gates into consideration. Our algorithm resynthesizes quantum circuits composed of Clifford group and T gates, the latter being typically the most costly gate in fault-tolerant models, e.g., those based on the Steane or surface codes, with the purpose of minimizing both T-count and T-depth. A major feature of the algorithm is the ability to resynthesize circuits with ancillae at effectively no additional cost, allowing space-time trade-offs to be easily explored. The tested benchmarks show up to 65.7% reduction in T-count and up to 87.6% reduction in T-depth without ancillae, or 99.7% reduction in T-depth using ancillae.

Verified Compilation of Space-Efficient Reversible Circuits

The generation of reversible circuits from high-level code is an important problem in several application domains, including low-power electronics and quantum computing. Existing tools compile and optimize reversible circuits for various metrics, such as the overall circuit size or the total amount of space required to implement a given function reversibly. However, little effort has been spent on verifying the correctness of the results, an issue of particular importance in quantum computing. There, compilation allows not only mapping to hardware, but also the estimation of resources required to implement a given quantum algorithm, a process that is crucial for identifying which algorithms will outperform their classical counterparts. We present a reversible circuit compiler called ReVerC, which has been formally verified in F* and compiles circuits that operate correctly with respect to the input program. Our compiler compiles the Revs language to combinational reversible circuits with as few ancillary bits as possible, and provably cleans temporary values.

Estimating the Cost of Generic Quantum Pre-image Attacks on SHA-2 and SHA-3

We investigate the cost of Grover’s quantum search algorithm when used in the context of pre-image attacks on the SHA-2 and SHA-3 families of hash functions. Our cost model assumes that the attack is run on a surface code based fault-tolerant quantum computer. Our estimates rely on a time-area metric that costs the number of logical qubits times the depth of the circuit in units of surface code cycles. As a surface code cycle involves a significant classical processing stage, our cost estimates allow for crude, but direct, comparisons of classical and quantum algorithms.

Complexity of reversible circuits and their quantum implementations

We provide an extensive overview of upper bounds on the number of gates needed in reversible and quantum circuits. As reversible gate libraries we consider single-target gates, mixed-polarity multiple-controlled Toffoli gates, and the set consisting of the NOT, the CNOT, and the two-controlled Toffoli gate. As quantum gate libraries we consider the semi-classical NCV library (consisting of NOT, CNOT, and the square-root of NOT called V) as well as the universal and commonly used Clifford + T gate library. Besides a summary of known bounds, the paper provides several new and tighter bounds. Several synthesis approaches and mapping schemes were used to calculate the bounds.

Technology Mapping of Reversible Circuits to Clifford+T Quantum Circuits

The Clifford+T quantum gate library has attracted much interest in the design of quantum circuits, particularly since the contained operations can be implemented in a fault-tolerant manner. Since fault tolerant implementations of the T gate have very high latency, synthesis and optimization are aiming at minimizing the number of T stages, referred to as the T-depth. In this paper, we present an approach to map mixed polarity multiple controlled Toffoli gates into Clifford+T quantum circuits. Our approach is based on the multiple control Toffoli mapping algorithms proposed by Barenco et al., which are given T-depth optimized Clifford+T translations. Experiments show that our approach leads to a significant T-depth reduction of 54% on average.

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

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%.We study the problem of -optimal quantum circuit synthesis over gate sets consisting of 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 cost of an implementation. As a corollary we prove that minimization over and phase gates is at least as hard as synthesizing a -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 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 optimization over more general Clifford+T circuits. We further present an efficient heuristic algorithm for synthesizing circuits over and Z-basis rotations with small cost. Our experiments show a 23% reduction of gates on average across a suite of Clifford+T benchmark circuits, with a maximum reduction of 43%.