Quantum algorithms for classical Boolean functions via adaptive measurements: Exponential reductions in space-time resources

The limited computational power of constant-depth quantum circuits can be boosted by adapting future gates according to the outcomes of mid-circuit measurements. We formulate computation of a variety of Boolean functions in the framework of adaptive measurement-based quantum computation using a cluster state resource and a classical side-processor that can add bits modulo 2, so-called $l2$-MBQC. Our adaptive approach overcomes a known challenge that computing these functions in the nonadaptive setting requires a resource state that is exponentially large in the size of the computational input. In particular, we construct adaptive $l2$-MBQC algorithms based on the quantum signal processing technique that compute the mod-$p$ functions with the best known scaling in the space-time resources (i.e., qubit count, quantum circuit depth, classical memory size, and number of calls to the side-processor). As the subject is diverse and has a long history, the paper includes reviews of several previously constructed algorithms and recasts them as adaptive $l2$-MBQCs using cluster state resources. Our results constitute an alternative proof of an old theorem regarding an oracular separation between the power of constant-depth quantum circuits and constant-depth classical circuits with unbounded fan-in NAND and mod-$p$ gates for any prime $p$.Comment: 15 + 14 pages, 4 figure

Automated Quantum Oracle Synthesis with a Minimal Number of Qubits

Several prominent quantum computing algorithms--including Grover's search algorithm and Shor's algorithm for finding the prime factorization of an integer--employ subcircuits termed 'oracles' that embed a specific instance of a mathematical function into a corresponding bijective function that is then realized as a quantum circuit representation. Designing oracles, and particularly, designing them to be optimized for a particular use case, can be a non-trivial task. For example, the challenge of implementing quantum circuits in the current era of NISQ-based quantum computers generally dictates that they should be designed with a minimal number of qubits, as larger qubit counts increase the likelihood that computations will fail due to one or more of the qubits decohering. However, some quantum circuits require that function domain values be preserved, which can preclude using the minimal number of qubits in the oracle circuit. Thus, quantum oracles must be designed with a particular application in mind. In this work, we present two methods for automatic quantum oracle synthesis. One of these methods uses a minimal number of qubits, while the other preserves the function domain values while also minimizing the overall required number of qubits. For each method, we describe known quantum circuit use cases, and illustrate implementation using an automated quantum compilation and optimization tool to synthesize oracles for a set of benchmark functions; we can then compare the methods with metrics including required qubit count and quantum circuit complexity.Comment: 18 pages, 10 figures, SPIE Defense + Commercial Sensing: Quantum Information Science, Sensing, and Computation X

Discrete Wigner functions and quantum computational speedup

In [Phys. Rev. A 70, 062101 (2004)] Gibbons et al. defined a class of discrete Wigner functions W to represent quantum states in a finite Hilbert space dimension d. I characterize a set C_d of states having non-negative W simultaneously in all definitions of W in this class. For d<6 I show C_d is the convex hull of stabilizer states. This supports the conjecture that negativity of W is necessary for exponential speedup in pure-state quantum computation.Comment: 7 pages, 2 figures, RevTeX. v2: clarified discussion on dynamics, added refs., published versio

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared in the Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199