Parallelizing quantum circuit synthesis

We present an algorithmic framework for parallel quantum circuit synthesis using meet-in-the-middle synthesis techniques. We also present two implementations thereof, using both threaded and hybrid parallelization techniques. We give examples where applying parallelism offers a speedup on the time of circuit synthesis for 2- and 3-qubit circuits. We use a threaded algorithm to synthesize 3-qubit circuits with optimal T -count 9, and 11, breaking the previous record of T-count 7. As the estimated runtime of the framework is inversely proportional to the number of processors, we propose an implementation using hybrid parallel programming which can take full advantage of a computing cluster’s thousands of cores. This implementation has the potential to synthesize circuits which were previously deemed impossible due to the exponential runtime of existing algorithms

Simple factorization of unitary transformations

We demonstrate a method for general linear optical networks that allows one to factorize any SU($n$) matrix in terms of two SU($n-1)$ blocks coupled by an SU(2) entangling beam splitter. The process can be recursively continued in an efficient way, ending in a tidy arrangement of SU(2) transformations. The method hinges only on a linear relationship between input and output states, and can thus be applied to a variety of scenarios, such as microwaves, acoustics, and quantum fields.Comment: 5 pages, 4 figures. Comments welcome

Discrete phase-space approach to mutually orthogonal Latin squares

We show there is a natural connection between Latin squares and commutative sets of monomials defining geometric structures in finite phase-space of prime power dimensions. A complete set of such monomials defines a mutually unbiased basis (MUB) and may be associated with a complete set of mutually orthogonal Latin squares (MOLS). We translate some possible operations on the monomial sets into isomorphisms of Latin squares, and find a general form of permutations that map between Latin squares corresponding to unitarily equivalent mutually unbiased sets. We extend this result to a conjecture: MOLS associated to unitarily equivalent MUBs will always be isomorphic, and MOLS associated to unitarily inequivalent MUBs will be non-isomorphic

Methods for parallel quantum circuit synthesis, fault-tolerant quantum RAM, and quantum state tomography

The pace of innovation in quantum information science has recently exploded due to the hope that a quantum computer will be able to solve a multitude of problems that are intractable using classical hardware. Current quantum devices are in what has been termed the ``noisy intermediate-scale quantum'', or NISQ stage. Quantum hardware available today with 50-100 physical qubits may be among the first to demonstrate a quantum advantage. However, there are many challenges to overcome, such as dealing with noise, lowering error rates, improving coherence times, and scalability. We are at a time in the field where minimization of resources is critical so that we can run our algorithms sooner rather than later. Running quantum algorithms ``at scale'' incurs a massive amount of resources, from the number of qubits required to the circuit depth. A large amount of this is due to the need to implement operations fault-tolerantly using error-correcting codes. For one, to run an algorithm we must be able to efficiently read in and output data. Fault-tolerantly implementing quantum memories may become an input bottleneck for quantum algorithms, including many which would otherwise yield massive improvements in algorithm complexity. We will also need efficient methods for tomography to characterize and verify our processes and outputs. Researchers will require tools to automate the design of large quantum algorithms, to compile, optimize, and verify their circuits, and to do so in a way that minimizes operations that are expensive in a fault-tolerant setting. Finally, we will also need overarching frameworks to characterize the resource requirements themselves. Such tools must be easily adaptable to new developments in the field, and allow users to explore tradeoffs between their parameters of interest. This thesis contains three contributions to this effort: improving circuit synthesis using large-scale parallelization; designing circuits for quantum random-access memories and analyzing various time/space tradeoffs; using the mathematical structure of discrete phase space to select subsets of tomographic measurements. For each topic the theoretical work is supplemented by a software package intended to allow others researchers to easily verify, use, and expand upon the techniques herein

Fault tolerant resource estimation of quantum random-access memories

Quantum random-access look-up of a string of classical bits is a necessary ingredient in several important quantum algorithms. In some cases, the cost of such quantum random-access memory (qRAM) is the limiting factor in the implementation of the algorithm. In this paper we study the cost of fault-tolerantly implementing a qRAM. We construct generic families of circuits which function as a qRAM, and analyze their resource costs when embedded in a surface code.Comment: 17 pages, 12 figures. Code repository available in reference

Exploring the Potential of Qutrits for Quantum Optimization of Graph Coloring

Recent hardware demonstrations and advances in circuit compilation have made quantum computing with higher-dimensional systems (qudits) on near-term devices an attractive possibility. Some problems have more natural or optimal encodings using qudits over qubits. We explore this potential by formulating graph 3-coloring, a well-known and difficult problem with practical applications, using qutrits, and solve it using the quantum approximate optimization algorithm (QAOA). Qutrit-based cost and mixer Hamiltonians are constructed along with appropriate quantum circuits using qutrit gates. We run noiseless simulations using PennyLane to compare the formulation against qubit-based QAOA, and analyze the solution quality and resources required. Preliminary results show that the qutrit encoding finds more accurate solutions with a comparable set of hyperparameters, uses half as many qudits, and has a notably smaller circuit depth per layer than an efficient qubit encoding. This work suggests that qutrits may be useful in solving some problems on near-term devices, however further work is required to assess their potential in a noisy environment.Comment: Accepted in IEEE QCE23 (New Ideas and Emergent Results