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arXiv:2305.01816 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Numerical Circuit Synthesis and Compilation for Multi-State Preparation

Aaron Szasz, Ed Younis, Wibe Albert de Jong

Published Sep 17, 2023• 4 citations
arXivHTMLPDF

Abstract

Near-term quantum computers have significant error rates and short coherence times, so compilation of circuits to be as short as possible is essential. Two types of compilation problems are typically considered: circuits to prepare a given state from a fixed input state, called “state preparation”; and circuits to implement a given unitary operation, for example by “unitary synthesis”. In this paper we solve a more general problem: the transformation of a set of m states to another set of$m$states, which we call “multi-state preparation”. State preparation and unitary synthesis are special cases; for state preparation, m=1, while for unitary synthesis, m is the dimension of the full Hilbert space. We generate and optimize circuits for multi-state preparation numerically. In cases where a top-down approach based on matrix decompositions is also possible, our method finds circuits with substantially (up to 40 %) fewer two-qubit gates. We discuss possible applications, including efficient preparation of macroscopic superposition (“cat”) states and synthesis of quantum channels.

DOI
10.1109/qce57702.2023.00092

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Near-term quantum computers have significant error rates and short coherence times, so compilation of circuits to be as short as possible is essential. Two types of compilation problems are typically considered: circuits to prepare a given state from a fixed input state, called “state preparation”; and circuits to implement a given unitary operation, for example by “unitary synthesis”. In this paper we solve a more general problem: the transformation of a set of m states to another set ofmmmstates, which we call “multi-state preparation”. State preparation and unitary synthesis are special cases; for state preparation, m=1, while for unitary synthesis, m is the dimension of the full Hilbert space. We generate and optimize circuits for multi-state preparation numerically. In cases where a top-down approach based on matrix decompositions is also possible, our method finds circuits with substantially (up to 40 %) fewer two-qubit gates. We discuss possible applications, including efficient preparation of macroscopic superposition (“cat”) states and synthesis of quantum channels.
We perform optimal-control-theory calculations to determine the minimum number of two-qubit controlled-not (cnot) gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that all produce the desired result, even at the minimum number of gates. We also show that the number of entangling gates can be reduced if we use multiqubit entangling gates instead of two-qubit cnot gates, as one might expect based on parameter counting calculations. In addition to treating the general case of arbitrary target states or unitary operators, we apply the numerical approach to the special case of synthesizing the multiqubit Toffoli gate. This approach can be used to investigate any other specific few-qubit task and provides insight into the tightness of different bounds in the literature.
The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2^n x 2^n unitary matrices into efficient circuits of (n-1)-controlled single-qubit and (n-1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of self-inverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an numerative algorithm for minimal (n-1)-controlled-NOT circuit construction, which we call the Palindromic Optimization Algorithm. Our work dramatically reduces the number of gates generated by the conventional two-level decomposition method for constructing quantum circuits of (n-1)-controlled single-qubit and (n-1)-controlled-NOT gates.
The efficiency of compiling high-level quantum algorithms into instruction sets native to quantum computers defines the moment in the future when we will be able to solve interesting and important problems on quantum computers. In my work I focus on the new methods for compiling single qubit operations that appear in many quantum algorithms into single qubit operations natively supported by several popular architectures. In addition, I study several questions related to synthesis and optimization of multiqubit operations. \n \n \nWhen studying the single qubit case, I consider two native instruction sets. The first one is Clifford+T; it is supported by conventional quantum computers implementing fault tolerance protocols based on concatenated and surface codes, and by topological quantum computers based on Ising anyons. The second instruction set is the one supported by topological quantum computers based on Fibonacci anyons. I show that in both cases one can use the number theoretic structure of the problem and methods of computational algebraic number theory to achieve improvements over the previous state of the art by factors ranging from 10 to 1000 for instances of the problem interesting in practice. This order of improvement might make certain interesting quantum computations possible several years earlier. \n \n \nThe work related to multiqubit operations is on exact synthesis and optimization of Clifford+T and Clifford circuits. I show an exact synthesis algorithm for unitaries generated by Clifford+T circuits requiring exponentially less number of gates than previous state of the art. For Clifford circuits two directions are studied: the algorithm for finding optimal circuits acting on a small number of qubits and heuristics for larger circuits optimization. The techniques developed allows one to reduce the size of encoding and decoding circuits for quantum error correcting codes by 40-50\\% and also finds their applications in randomized benchmarking protocols.
Abstract As quantum processors grow in scale and reliability, the need for efficient quantum gate decomposition of circuits to a set of specific available gates, becomes ever more critical. The decomposition of a particular algorithm into a sequence of these available gates is not unique. Thus, the fidelity of an algorithm’s implementation can be increased by choosing an optimized decomposition. This is true both for noisy intermediate-scale quantum platforms as well as for implementation of quantum error correction schemes. Here we present a compilation scheme which implements a general-circuit decomposition to a sequence of Ising-type, long-range, multi-qubit (MQ) entangling gates, that are separated by layers of single qubit rotations. We use trapped ions as an example in which MQ gates naturally arise, yet any system that has connectivity beyond nearest-neighbors may gain from our approach. We evaluate our methods using the quantum volume (QV) test over N qubits. In this context, our method replaces 3 N 2 / 2 two-qubit gates with 2 N + 1 MQ gates. Furthermore, our method minimizes the magnitude of the entanglement phases, which typically enables an improved implementation fidelity, by using weaker driving fields or faster realizations. We numerically test our compilation and show that, compared to conventional realizations with sequential two-qubit gates, our compilations improves the logarithm of QV by 20% to 25%.
Quantum circuit synthesis is the process in which an arbitrary unitary operation is decomposed into a sequence of gates from a universal set, typically one which a quantum computer can implement both efficiently and fault-tolerantly. As physical implementations of quantum computers improve, the need is growing for tools that can effectively synthesize components of the circuits and algorithms they will run. Existing algorithms for exact, multi-qubit circuit synthesis scale exponentially in the number of qubits and circuit depth, leaving synthesis intractable for circuits on more than a handful of qubits. Even modest improvements in circuit synthesis procedures may lead to significant advances, pushing forward the boundaries of not only the size of solvable circuit synthesis problems, but also in what can be realized physically as a result of having more efficient circuits. We present a method for quantum circuit synthesis using deterministic walks. Also termed pseudorandom walks, these are walks in which once a starting point is chosen, its path is completely determined. We apply our method to construct a parallel framework for circuit synthesis, and implement one such version performing optimal T -count synthesis over the Clifford+ T gate set. We use our software to present examples where parallelization offers a significant speedup on the runtime, as well as directly confirm that the 4-qubit 1-bit full adder has optimal T -count 7 and T -depth 3.
To implement quantum algorithms on quantum computers it is crucial to decompose their operators into the limited gate set supported by those computers. Unfortunately, existing works automating this essential task are generally slow and only applicable to narrow use cases.We present Synthetiq, a method to synthesize quantum circuits implementing a given specification over arbitrary finite gate sets, which is faster and more versatile than existing works. Synthetiq utilizes Simulated Annealing instantiated with a novel, domain-specific energy function that allows developers to leverage partial specifications for better efficiency. Synthetiq further couples this synthesis method with a custom simplification pass, to ensure efficiency of the found circuits. We experimentally demonstrate that Synthetiq can generate better implementations than were previously known for multiple relevant quantum operators including RCCCX, CCT, CCiSWAP, C√SWAP, and C√iSWAP. Our extensive evaluation also demonstrates Synthetiq frequently outperforms a wide variety of more specialized tools in their own domains, including (i) ‍the well-studied task of synthesizing fully specified operators in the Clifford+T gate set, (ii) ‍є-approximate synthesis of multi-qubit operators in the same gate set, and (iii) ‍synthesis tasks with custom gate sets. On all those tasks, Synthetiq is typically one to two orders of magnitude faster than previous state-of-the-art and can tackle problems that were previously out of the reach of any synthesis tool.