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arXiv:2204.13524 · Physical Review A

Numerical analysis of quantum circuits for state preparation and unitary operator synthesis

Sahel Ashhab, Naoki Yamamoto, Fumiki Yoshihara, Kouichi Semba

Published Aug 23, 2022• 20 citations
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Abstract

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.

DOI
10.1103/physreva.106.022426

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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.
We use a random search technique to find quantum gate sequences that implement perfect quantum state preparation or unitary operator synthesis with arbitrary targets. This approach is based on the recent discovery that there is a large multiplicity of quantum circuits that achieve unit fidelity in performing a given target operation, even at the minimum number of single-qubit and two-qubit gates needed to achieve unit fidelity. We show that the fraction of perfect-fidelity quantum circuits increases rapidly as soon as the circuit size exceeds the minimum circuit size required for achieving unit fidelity. This result implies that near-optimal quantum circuits for a variety of quantum information processing tasks can be identified relatively easily by trying only a few randomly chosen quantum circuits and optimizing their parameters. In addition to analyzing the case where the cnot gate is the elementary two-qubit gate, we consider the possibility of using alternative two-qubit gates. In particular, we analyze the case where the two-qubit gate is the BBB gate, which is known to reduce the minimum quantum circuit size for two-qubit operations. We apply the random search method to the problem of decomposing the four-qubit Toffoli gate and find a 15-cnot-gate decomposition.
We analyze the complexity of synthesizing random states and unitary operators in a multi-qudit system in two paradigms. In one case, we consider the situation in which we manipulate the system by applying a sequence of one- and two-qudit quantum gates that constitute the elementary, and universal, gate set. The minimum number of gates required to perform the desired operation represents the computational complexity. In the other case, we consider the situation in which we manipulate the physical system using physical fields with optimized control pulses. The minimum time required to perform the desired operation represents the physical complexity. In both cases, we use analytical arguments in combination with optimal-control-theory numerical calculations to determine the complexity of random operations. We show that the computational complexity of random states or unitary operators scales exponentially with the number of qudits. Our numerical results suggest that the physical complexity of preparing random quantum states and unitary operators scales more slowly than the computational complexity. We discuss various implications of our results, especially concerning the relationship between random and pseudorandom states and unitary operators.
Although with some yet severe limitations, Physical working Quantum Computers are becoming available for end users. Such devices are based on the rules of Quantum Mechanics, which state that physical systems evolve through unitary transformations. However, as in the classical case, in order to have a model of computation, such unitary evolutions are expressed/approximated inside Quantum Computers in terms of a finite set of one/two qubit operations (i.e., through a universal set of gates). Single qubit gates are fault-tolerant, while the same cannot be said for two qubit gates. Hence, unitary matrices adopted in Quantum Algorithms must be synthesized in terms of this universal set of operations to obtain a quantum circuit. In such synthesis procedure we prefer circuits with minimum number of qubits and with minimum circuit depth. Clifford+T universal set is one of the most adopted in the literature for synthesis. In such set we have 3 single qubit gates and the CNOT, which is a two qubit gate. Many efforts have been directed to devise algorithms that synthesize general unitary matrices into Clifford+T circuits. These algorithms usually tend to optimize circuit depth or eventually the number of T gates. Since two qubit gates are not fault tolerant, in this work we propose an ASP based technique to minimize the number of CNOT gates inside a Clifford+T circuit. Moreover, in real world quantum computers, qubit are usually connected to each other according to some particular topology, thus providing further constraints. Two qubit gates —hence, CNOT gate— can be directly applied only to pair of gates that are connected. Such constraint has to be taken into account during the synthesis of CNOT minimal circuits. We propose an ASP model to solve the problem of synthesizing CNOT minimal circuits under topological constraints. We provide experimental evidence of the scalability of our proposal.
State-of-the-art studies on quantum layout synthesis have proposed various approaches based on the assumption that the input circuit is only composed of single-qubit and two-qubit gates. This assumption greatly simplifies the layout synthesis problem, and thus they only require ensuring that all controlled-NOT (CNOT) gates satisfy the hardware constraints imposed by a given coupling graph. However, during the design of quantum circuits, multi-controlled Toffoli (MCT) gates are usually used to better characterize the function of the circuits. Directly decomposing them into single and two-qubit gates with a fixed routine ignores the flexibility and optimization opportunity provided by various decomposition (i.e., logic synthesis) possibilities and thus suffers from sub-optimal results. This article proposes a co-optimization approach for quantum logic and layout synthesis. The MCT gates are first decomposed into Toffoli gates, and an efficient qubit mapping checking process is proposed to optimally solve the SWAP-free layout synthesis problem by automatically determining the decomposition result of each Toffoli gate. If a SWAP-free result cannot be found, the proposed algorithm flow is then used to obtain a solution that minimizes the total cost that simultaneously counts the cost caused by the different decomposition methods for Toffoli gates and the cost induced by SWAP gates. Compared with a state-of-the-art method, the proposed approach reduces the number of additional CNOT gates by 16% with around 25X runtime speedup for the cases with SWAP-free solutions, which cannot be obtained without the co-optimization approach.
We show that the problem of synthesizing a quantum circuit for a given Clifford operator using at most a specified amount of resources is NP-complete for several resource models of practical interest. Specifically, we study synthesis of a given Clifford operator using single-qubit Clifford gates and two-qubit Clifford gates, with full qubit connectivity and no additional ancilla qubits. With single-qubit gates free and CNOT as the only two-qubit gate, we show that deciding whether a given target admits an implementation using at most a specified number of CNOT gates is NP-complete, and we show that the problem remains NP-complete when any two-qubit Clifford gate is allowed and the budget counts all such gates. Moreover, we show that both results continue to hold when SWAP gates may be inserted for free, and when single-qubit gates are also considered part of the budget. In each case, we build on recent work by Li and Guan and show that there are polynomial-time reductions from Hamiltonian path in grid graphs to these decision problems, using a number of qubits quadratic in the number of grid vertices.