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IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

Quantum Circuit Design for Integer Multiplication Based on Schönhage–Strassen Algorithm

Junhong Nie, Qinlin Zhu, Meng Li, Xiaoming Sun

Published May 30, 2023• 16 citations
Publisher

Abstract

Quantum arithmetic circuits have attracted extensive attention recently since it plays fundamental roles in many applications of quantum computing. Specifically, quantum circuits for integer multiplication are of great significance to various quantum algorithms, including Shor’s integer factorization and discrete logarithm algorithm. In this article, we design a family of quantum circuits for integer multiplication based on the famous classical integer multiplication algorithm, Schönhage–Strassen algorithm. We have made slight modifications to the algorithm to simplify its quantum circuit implementation. As a result, the quantum circuit we designed has gate depth$O(\log ^{2} n)$. To the best of our knowledge, this is the first poly-logarithmic depth quantum circuit for integer multiplication which keeps the circuit size and the number of ancillary qubits subquadratic. Our design has size$O(n\log n\log \log n)$counted by elementary quantum gates which is the same as the time complexity of the Schönhage–Strassen algorithm, and it consumes$O(n\log n\log \log n)$clean ancillary qubits. In addition, we also utilize a weaker version of Schönhage–Strassen algorithm to give a family of circuits which has depth at the same order$O(\log ^{2} n)$but with significantly smaller constants, while still keeping the size and number of ancillary qubits subquadratic.

DOI
10.1109/tcad.2023.3279300

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Quantum arithmetic circuits have attracted extensive attention recently since it plays fundamental roles in many applications of quantum computing. Specifically, quantum circuits for integer multiplication are of great significance to various quantum algorithms, including Shor’s integer factorization and discrete logarithm algorithm. In this article, we design a family of quantum circuits for integer multiplication based on the famous classical integer multiplication algorithm, Schönhage–Strassen algorithm. We have made slight modifications to the algorithm to simplify its quantum circuit implementation. As a result, the quantum circuit we designed has gate depthO(log⁡2n)O(\log ^{2} n)O(log2n). To the best of our knowledge, this is the first poly-logarithmic depth quantum circuit for integer multiplication which keeps the circuit size and the number of ancillary qubits subquadratic. Our design has sizeO(nlog⁡nlog⁡log⁡n)O(n\log n\log \log n)O(nlognloglogn)counted by elementary quantum gates which is the same as the time complexity of the Schönhage–Strassen algorithm, and it consumesO(nlog⁡nlog⁡log⁡n)O(n\log n\log \log n)O(nlognloglogn)clean ancillary qubits. In addition, we also utilize a weaker version of Schönhage–Strassen algorithm to give a family of circuits which has depth at the same orderO(log⁡2n)O(\log ^{2} n)O(log2n)but with significantly smaller constants, while still keeping the size and number of ancillary qubits subquadratic.
Multiplication over binary fields is a crucial operation in quantum algorithms designed to solve the discrete logarithm problem for elliptic curve defined over GF(2n)GF(2^n)GF(2n). In this paper, we present an algorithm for constructing quantum circuits that perform multiplication over GF(2n)GF(2^n)GF(2n) with O(nlog⁡2(3))\mathcal{O}(n^{\log_2(3)})O(nlog2​(3)) Toffoli gates. We propose a variant of our construction that achieves linear depth by using O(nlog⁡2(n))\mathcal{O}(n\log_2(n))O(nlog2​(n)) ancillary qubits. This approach provides the best known space-time trade-off for binary field multiplication with a subquadratic number of Toffoli gates. Additionally, we demonstrate that for some particular families of primitive polynomials, such as trinomials, the multiplication can be done in logarithmic depth and with O(nlog⁡2(3))\mathcal{O}(n^{\log_2(3)})O(nlog2​(3)) gates.
The multiplication of superpositions of numbers is a core operation in many quantum algorithms. The standard method for multiplication (both classical and quantum) has a runtime quadratic in the size of the inputs. Quantum circuits with asymptotically fewer gates have been developed, but generally exhibit large overheads, especially in the number of ancilla qubits. In this work, we introduce a new paradigm for sub-quadratic-time quantum multiplication with zero ancilla qubits -- the only qubits involved are the input and output registers themselves. Our algorithm achieves an asymptotic gate count of O(n1+ε)\mathcal{O}(n^{1+ε})O(n1+ε) for any ε>0ε> 0ε>0; with practical choices of parameters, we expect scalings as low as O(n1.3)\mathcal{O}(n^{1.3})O(n1.3). Used as a subroutine in Shor's algorithm, our technique immediately yields a factoring circuit with O(n2+ε)\mathcal{O}(n^{2+ε})O(n2+ε) gates and only 2n+O(log⁡n)2n + \mathcal{O}(\log n)2n+O(logn) qubits; to our knowledge, this is by far the best qubit count of any factoring circuit with a sub-cubic number of gates. Used in Regev's recent factoring algorithm, the gate count is O(n1.5+ε)\mathcal{O}(n^{1.5+ε})O(n1.5+ε). Finally, we demonstrate that our algorithm has the potential to outperform previous proposals at problem sizes relevant in practice, including yielding the smallest circuits we know of for classically-verifiable quantum advantage.
Optimizing the depth and number of ancillary qubits in quantum circuits is crucial in quantum computation, given the limitations imposed by current quantum devices. In this article, we introduce an innovative approach for implementing arbitrary symmetric Boolean functions using poly-logarithmic depth quantum circuits with only a logarithmic number of ancillary qubits. Symmetric functions are those whose outputs are dictated solely by the Hamming weight of the inputs. These functions find applications across various domains, including quantum machine learning and arithmetic circuit synthesis. Moreover, by fully leveraging the potential of qutrits, the ancilla count can be further reduced to just one. The key technique involves a novel poly-logarithmic depth quantum circuit designed to compute Hamming weight without the need for ancillary qubits. This quantum circuit for Hamming weight is of independent interest due to its wide-ranging applications, such as in quantum memory, quantum machine learning, and Hamiltonian dynamics simulations.
Quantum computers offer the potential to extend our abilities to tackle computational problems in fields such as number theory, encryption, search and scientific computation. Up to a superpolynomial speedup has been reported for quantum algorithms in these areas. Motivated by the promise of faster computations, the development of quantum machines has caught the attention of both academics and industry researchers. Quantum machines are now at sizes where implementations of quantum algorithms or their components are now becoming possible. In order to implement quantum algorithms on quantum machines, resource efficient circuits and functional blocks must be designed. In this work, we propose quantum circuits for Galois and integer arithmetic. These quantum circuits are necessary building blocks to realize quantum algorithms. The design of resource efficient quantum circuits requires the designer takes into account the gate cost, quantum bit (qubit) cost, depth and garbage outputs of a quantum circuit. Existing quantum machines do not have many qubits meaning that circuits with high qubit cost cannot be implemented. In addition, quantum circuits are more prone to errors and garbage output removal adds to overall cost. As more gates are used, a quantum circuit sees an increased rate of failure. Failures and error rates can be countered by using quantum error correcting codes and fault tolerant implementations of universal gate sets (such as Clifford+T gates). However, Clifford+T gates are costly to implement with the T gate being significantly more costly than the Clifford gates. As a result, designers working with Clifford+T gates seek to minimize the number of T gates (T-count) and the depth of T gates (T-depth). In this work, we propose quantum circuits for Galois and integer arithmetic with lower T-count, T-depth and qubit cost than existing work. This work presents novel quantum circuits for squaring and exponentiation over binary extension fields (Galois fields of form GF(2 m )). The proposed circuits are shown to have lower depth, qubit and gate cost to existing work. We also present quantum circuits for the core operations of multiplication and division which enjoy lower T-count, T-depth and qubit costs compared to existing work. This work also illustrates the design of a T-count and qubit cost efficient design for the square root. This work concludes with an illustration of how the arithmetic circuits can be combined into a functional block to implement quantum image processing algorithms.
Quantum circuits for multiplication are necessary for scientific computing on quantum computers. Clifford + T circuits are widely used in fault-tolerant implementations. The costs of implementing T and double-qubit gates are higher than those of other one-qubit gates in the Clifford + T group. In addition, the small number of qubits available in existing quantum devices is another constraint on quantum circuits. Therefore, we reduce T and CNOT gates and circuit width as the primary optimization goal in this paper. We propose an algorithm for multiplication with two unchanged operands. Preserving both operands of the multiplication is important for realizing some quantum algorithms, such as quantum bilinear interpolation. Using this algorithm, we design the circuit of the integer multiplier with two unchanged operands reducing CNOT gates. Next, we develop a Clifford + T circuit for the multiplier and introduce new optimization rules to reduce T gates. Comparative analysis shows that the proposed multiplier achieves the best width among existing multipliers. Compared to multipliers with two unchanged operands that use at most one ancillary qubit, our proposed multiplier has the best T-count, T-depth, and CNOT-count.
Quantum circuits performing arithmetic operations are critical in quantum computing because of the need for such operations in proven quantum algorithms. Although quantum computers are becoming increasingly resourceful, the number of qubits currently available is still limited. Furthermore, these qubits are heavily affected by internal and external noise. It has been proven that quantum circuits built using Clifford +T+T+T gates can be made fault tolerant. However, the use of the TTT gates comes at a very high cost. If the number of TTT gates used in a circuit is not optimized, the cost of the circuit will be increased excessively. As a consequence, it is essential to optimize the circuits so that they are as resource efficient as possible and also noise tolerant. This paper presents the design of a circuit to perform the multiplication of two integers. The circuit is built using only Clifford +T+T+T gates for compatibility with error detection and correction codes. It outperforms the circuits in the state of the art in terms of TTT count and TTT depth.