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IEEE Symposium on Computer Arithmetic (ARITH)

Towards Quantum Logarithm Number Systems

Mark G. Arnold

Published Sep 1, 2022• 1 citation
Publisher

Abstract

Quantum computers, which process qubits, offer the promise of spectacular performance improvement over ordinary computers that deal only with classical bits, but there are obstacles to this vision. First, current quantum technology only allows a small number of qubits, and these are susceptible to noise. Second, quantum algorithms must be reversible, which often requires ancillary data that consume precious qubits. Third, interesting algorithms amenable to quantum implementation, such as chemistry simulation, require representing real numbers. Although quantum integer arithmetic has been studied extensively, the few works on quantum floating point demand more ancillary qubits than input data making floating point impractical for current quantum hardware. This paper suggests an alternative to floating point, known as the Logarithmic Number System (LNS), which has proven effective for approximate arithmetic with classical hardware. Reversible LNS multiplication and division are easy and exact with one ancillary qubit. Here we explore the quantum cost of difficult LNS operations (addition and subtraction). LNS offers implementation tradeoffs between accuracy and qubit cost that suggest highly-approximate LNS will be practical on quantum hardware sooner than when quantum technology has improved enough for floating-point to be practical.

DOI
10.1109/arith54963.2022.00022

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Quantum computers, which process qubits, offer the promise of spectacular performance improvement over ordinary computers that deal only with classical bits, but there are obstacles to this vision. First, current quantum technology only allows a small number of qubits, and these are susceptible to noise. Second, quantum algorithms must be reversible, which often requires ancillary data that consume precious qubits. Third, interesting algorithms amenable to quantum implementation, such as chemistry simulation, require representing real numbers. Although quantum integer arithmetic has been studied extensively, the few works on quantum floating point demand more ancillary qubits than input data making floating point impractical for current quantum hardware. This paper suggests an alternative to floating point, known as the Logarithmic Number System (LNS), which has proven effective for approximate arithmetic with classical hardware. Reversible LNS multiplication and division are easy and exact with one ancillary qubit. Here we explore the quantum cost of difficult LNS operations (addition and subtraction). LNS offers implementation tradeoffs between accuracy and qubit cost that suggest highly-approximate LNS will be practical on quantum hardware sooner than when quantum technology has improved enough for floating-point to be practical.
We propose a novel floating-point encoding scheme that builds on prior work involving fixed-point encodings. We encode floating-point numbers using Two's Complement fixed-point mantissas and Two's Complement integral exponents. We used our proposed approach to develop quantum algorithms for fundamental arithmetic operations, such as bit-shifting, reciprocation, multiplication, and addition. We prototyped and investigated the performance of the floating-point encoding scheme on quantum computer simulations by performing reciprocation on randomly drawn inputs and by solving first-order ordinary differential equations, while varying the number of qubits in the encoding. We observed rapid convergence to the exact solutions as we increased the number of qubits and a significant reduction in the number of ancilla qubits required for reciprocation when compared with similar approaches.
There are several possible ways to encode random variables in a quantum state. The basis encoding of bit strings has paramount importance because it allows to load the values of a random variable through the superposition of corresponding basis states, and to then exploit quantum parallelism in processing algorithms. The basis encoding offers a natural way to represent an unsigned integer random variable, and extends to signed integers, as well as to fixed-point and floating-point variables. Each quantum representation of fractional numbers, however, involves a trade-off between accuracy and depth of manipulation circuits. Here, an efficient hybrid quantum-classical representation of quantum floating points is introduced. It combines a quantum register containing the values, with a classical register storing global information about the variable, namely the range and approximation tolerances. The sum and product operations are defined, in such a way as to ensure they are performed without overflow. By taking advantage of the stored classical information, the precision degradation that occurs due to rounding after repeated data manipulations, can be significantly reduced compared to known strategies. Ad hoc examples show up to around 90%90\%90% reduction in approximation, compared to previous techniques, after repeated additions. The method finds application in many algorithms of practical relevance and constitutes a significant advance in the design of arithmetic circuits with low depth and high accuracy.
Implementation of floating-point arithmetic with consistent rounding is a critical component of many quantum algorithms. Quantum circuit implementations for squaring and division serve as examples here. This work was motivated by ongoing work in developing quantum algorithms for scientific and engineering computing applications, where this type of arithmetic often forms part of the algorithm. A key feature of the work is the use of a reduced-precision floating-point representation of real data specifically designed for near-term future quantum computing hardware with a limited number of qubits (e.g., less than 100) and with an increased level of fault tolerance as compared to current quantum computing hardware. The quantum circuit implementations of the squaring of a floating-point number and the division of two floating-point numbers are detailed here, highlighting similarities in the quantum circuit implementation for the logical steps required for rounding-to-nearest in line with the IEEE 754 standard for the two arithmetic operations. This similarity is an important feature regarding future work where an automated generation of this type of quantum circuit from a set of standard modules and circuit templates is employed.
The Quantum Fourier Transform (QFT) grants competitive advantages, especially in resource usage and circuit approximation, for performing arithmetic operations on quantum computers, and offers a potential route towards a numerical quantum-computational paradigm. In this paper, we utilize efficient techniques to implement QFT-based integer addition and multiplications. These operations are fundamental to various quantum applications including Shor's algorithm, weighted sum optimization problems in data processing and machine learning, and quantum algorithms requiring inner products. We carry out performance evaluations of these implementations based on IBM's superconducting qubit architecture using different compatible noise models. We isolate the sensitivity of the component quantum circuits on both one-/two-qubit gate error rates, and the number of the arithmetic operands' superposed integer states. We analyze performance, and identify the most effective approximation depths for quantum add and quantum multiply within the given context. We observe significant dependency of the optimal approximation depth on the degree of machine noise and the number of superposed states in certain performance regimes. Finally, we elaborate on the algorithmic challenges - relevant to signed, unsigned, modular and non-modular versions - that could also be applied to current implementations of QFT-based subtraction, division, exponentiation, and their potential tensor extensions. We analyze performance trends in our results and speculate on possible future development within this computational paradigm.
Quantum algorithms for scientific computing require modules implementing fundamental functions, such as the square root, the logarithm, and others. We require algorithms that have a well-controlled numerical error, that are uniformly scalable and reversible (unitary), and that can be implemented efficiently. We present quantum algorithms and circuits for computing the square root, the natural logarithm, and arbitrary fractional powers. We provide performance guarantees in terms of their worst-case accuracy and cost. We further illustrate their performance by providing tests comparing them to the respective floating point implementations found in widely used numerical software.
Abstract The Quantum Fourier Transform (QFT) grants competitive advantages, especially in resource usage and circuit approximation, for performing arithmetic operations on quantum computers , and offers a potential route towards a numerical quantum-computational paradigm. In this paper, we utilize efficient techniques to implement QFT-based integer addition and multiplications. These operations are fundamental to various quantum applications including Shor’s algorithm, weighted sum optimization problems in data processing and machine learning, and quantum algorithms requiring inner products. We carry out performance evaluations of these implementations based on IBM’s superconducting qubit architecture using different compatible noise models. We isolate the sensitivity of the component quantum circuits on both one-/two-qubit gate error rates, and the number of the arithmetic operands’ superposed integer states. We analyze performance, and identify the most effective approximation depths for unsigned quantum addition and quantum multiplication within the given context. We then perform a similar analysis of signed addition and compare to the unsigned results. We observe significant dependency of the optimal approximation depth on the degree of machine noise and the number of superposed states in certain performance regimes. Finally, we elaborate on the algorithmic challenges - relevant to signed, unsigned, modular and non-modular versions - that could also be applied to current implementations of QFT-based subtraction, division, exponentiation, and their potential tensor extensions. We analyze the performance trends in our results and speculate on possible future developments within this computational paradigm.