In classical computing, 2’s complement representation is essential for handling signed binary arithmetic, enabling efficient operations for both positive and negative numbers. However, many quantum multiplication circuits typically lack native support for 2’s complement, limiting their application to signed arithmetic tasks. Prior quantum multiplication algorithms capable of handling 2’s complement often demand an impractically large number of qubits. This paper introduces 3 novel quantum circuit designs for real-number 2’s complement multiplication, inspired by classical signed multiplication algorithms. These designs achieve a quadratic time complexity, comparable to RCA-based quantum unsigned multiplication algorithms, while significantly reducing qubit requirements compared to prior quantum signed multiplication schemes. The first model requires the largest number of qubits but applies to a broad range of cases; the second uses fewer qubits but requires specific multiplier conditions; and the third uses the fewest qubits but operates under stricter multiplier constraints and circuit positioning conditions. Benchmarking results demonstrate their ability to perform 2’s complement multiplication correctly. This work advances qubit-efficient solutions, laying a foundation for further development of 2’s complement-based quantum algorithms.
Quantum circuits which perform integer arithmetic could potentially outperform their classical counterparts. In this paper, a quantum circuit is considered which performs a specific computational pattern on classically represented integers to accelerate the computation. Such a hybrid circuit could be embedded in a conventional computer architecture as a quantum device or accelerator. In particular, a quantum multiply-add circuit (QMAC) using a Quantum Fourier Transform (QFT) is proposed which can perform the calculation on conventional integers faster than its conventional counterpart. Whereas classically applying a multiply-adder (MAC)
n times to
k bit integers would require
O(nlogk) parallel steps, the hybrid QMAC needs only
O(n+k) steps for the exact result and
O(n+logk) steps for an approximate result.
This paper proposes a new and improved implementation of a quantum integer multiplier. Performing arithmetic computations is sometimes a necessary step in the implementation of quantum algorithms. In this work, Quantum Fourier Transform is used in order to perform scalable arithmetic in a generic bit-width quantum system. In the phase domain, addition can be implemented through accumulated controlled rotations on the qubits’ state. Leveraging this, and inspired by the classical implementation of an array multiplier, a new integer multiplier is fully designed and tested in a quantum environment.The depth of a quantum circuit is the number of computational steps necessary to completion, and it is a key parameter that reflects on the performance of the design. The new design reduces the quantum depth of the design from the exponential order of the previously proposed designs to polynomial order.
Quantum modular multipliers are a promising development in the field of quantum computing that aim to perform modular multiplication faster and more efficiently than classical algorithms. This is achieved by encoding the numbers being multiplied into quantum bits (qubits) and using quantum gates to manipulate them in a way that represents the calculation. Since quantum physics principles like superposition and entanglement dramatically speed up processing compared to conventional computing, many calculations may be run simultaneously. Quantum modular multipliers have the potential to have a significant impact in a range of fields, including cryptography and beyond, since they offer a faster and more efficient way to do modular multiplications. Several fields, including simulation, optimization, and cryptography, can benefit from the use of quantum computing.One of the crucial operations in many of these applications is modular multiplication, a sort of multiplication that requires obtaining the remainder of the product of two numbers when divided by a third number. In this paper, we provide a novel modular multiplication method based on a quantum algorithm. To examine its complexity and compare it to that of current adders, we employ a novel form of adder that is intended to not retain the final carry. This adder's decrease percentage in terms of gates and depth is roughly 70%. Our method takes advantage of the special qualities of qubits to complete the computation significantly more quickly than conventional algorithms. It is based on the fundamental ideas of quantum physics, such as superposition and entanglement.
Multiplication is a frequent computation in many algorithms, classical and quantum. This paper targets the im-plementation of quantum integer multiplication. Quantum array multipliers take inspiration from classical array multipliers, with the result of reduced circuit depth. They take advantage of the quantum phase domain, through rotations controlled by the mul-tiplier's qubits. This work further explores this implementation by applying approximate rotations. Although this approach can have an impact on the accuracy of the result, the reduction in depth can result in better outcomes when noise is involved.
Quantum multiplication is a fundamental operation in quantum computing. It is important to have a quantum multiplier with low complexity. In this paper, we propose the Quantum Multiplier Based on Exponent Adder (QMbead), a new approach that requires just
log2(n) qubits to multiply two
n-bit integer numbers, in addition to
O(n) ancillary qubits used for quantum state preparation. The QMbead uses a so-called exponent encoding to respectively represent two multiplicands as two superposition states which are prepared by a quantum state preparation method, then employs a quantum adder to obtain the sum of these two superposition states, and subsequently measures the outputs of the quantum adder to calculate the product of the multiplicands. Different quantum adders can be used in the QMbead. The circuit depth and time complexity of the QMbead, using a logarithmic-depth quantum carry lookahead adder (QCLA) as adder, are
O(logn) and
O(nlogn), respectively. The gate complexity of the QMbead is
O(n). The circuit depth and gate complexity of the QMbead is better than existing quantum multipliers such as the quantum Karatsuba multiplier and the QFT based multiplier. The time complexity of the QMbead is identical to that of the fastest classical multiplication algorithm, Harvey-Hoeven algorithm. Interestingly, the QMbead maintains an advantage over the Harvey-Hoeven algorithm, given that the latter is only suitable for excessively large numbers, whereas the QMbead is valid for both small and large numbers. The multiplicand can be either an integer or a decimal number. The QMbead has been implemented on quantum simulators to compute products with a bit length of up to 273 bits using only 17 qubits, excluding the ancillary qubits used for quantum state preparation. This establishes QMbead as an efficient solution for multiplying large integer or decimal numbers with many bits.