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Efficient Floating Point Arithmetic for Quantum Computers
Quantum Physics · Submitted 20 Dec 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The application of our methods to a 32-bit unsigned integer multiplication demonstrated a 90\% circuit depth reduction compared to carry-ripple approaches.
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The abstract, sentence by sentence
One of the major promises of quantum computing is the realization of SIMD (single instruction - multiple data) operations using the phenomenon of superposition. Since the dimension of the state space grows exponentially with the number of qubits, we can easily reach situations where we pay less than a single quantum gate per data point for data-processing instructions which would be rather expensive in classical computing. Formulating such instructions in terms of quantum gates, however, still remains a challenging task. Laying out the foundational functions for more advanced data-processing is therefore a subject of paramount importance for advancing the realm of quantum computing. In this paper, we introduce the formalism of encoding so called-semi-boolean polynomials. As it turns out, arithmetic $\mathbb{Z}/2^n\mathbb{Z}$ ring operations can be formulated as semi-boolean polynomial evaluations, which allows convenient generation of unsigned integer arithmetic quantum circuits. For arithmetic evaluations, the resulting algorithm has been known as Fourier-arithmetic. We extend this type of algorithm with additional features, such as ancilla-free in-place multiplication and integer coefficient polynomial evaluation. Furthermore, we introduce a tailor-made method for encoding signed integers succeeded by an encoding for arbitrary floating-point numbers. This representation of floating-point numbers and their processing can be applied to any quantum algorithm that performs unsigned modular integer arithmetic. We discuss some further performance enhancements of the semi boolean polynomial encoder and finally supply a complexity estimation. The application of our methods to a 32-bit unsigned integer multiplication demonstrated a 90\% circuit depth reduction compared to carry-ripple approaches.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | One of the major promises of quantum computing is the realization of SIMD (single instruction - multiple data) operations using the phenomenon of superposition. | background 1.00 | 0.06 | 0.03 | asserted 0.62 |
| s1 | Since the dimension of the state space grows exponentially with the number of qubits, we can easily reach situations where we pay less than a single quantum gate per data point for data-processing instructions which would be rather expensive in classical computing. | background 0.99 | 0.11 | 0.04 | asserted 0.64 |
| s2 | Formulating such instructions in terms of quantum gates, however, still remains a challenging task. | background 0.58 | 0.05 | 0.02 | asserted 1.00 |
| s3 | Laying out the foundational functions for more advanced data-processing is therefore a subject of paramount importance for advancing the realm of quantum computing. | background 0.77 | 0.05 | 0.03 | asserted 0.99 |
| s4 | In this paper, we introduce the formalism of encoding so called-semi-boolean polynomials. | contribution 0.96 | 0.92 | 0.07 | asserted 0.99 |
| s5 | As it turns out, arithmetic $\mathbb{Z}/2^n\mathbb{Z}$ ring operations can be formulated as semi-boolean polynomial evaluations, which allows convenient generation of unsigned integer arithmetic quantum circuits. | method 0.52 | 0.38 | 0.17 | asserted 0.99 |
| s6 | For arithmetic evaluations, the resulting algorithm has been known as Fourier-arithmetic. | background 0.99 | 0.06 | 0.04 | asserted 1.00 |
| s7 | We extend this type of algorithm with additional features, such as ancilla-free in-place multiplication and integer coefficient polynomial evaluation. | contribution 0.97 | 0.93 | 0.08 | asserted 1.00 |
| s8 | Furthermore, we introduce a tailor-made method for encoding signed integers succeeded by an encoding for arbitrary floating-point numbers. | contribution 0.81 | 0.94 | 0.10 | asserted 1.00 |
| s9 | This representation of floating-point numbers and their processing can be applied to any quantum algorithm that performs unsigned modular integer arithmetic. | interpretation 0.89 | 0.63 | 0.16 | asserted 0.93 |
| s10 | We discuss some further performance enhancements of the semi boolean polynomial encoder and finally supply a complexity estimation. | contribution 0.95 | 0.86 | 0.07 | asserted 0.77 |
| s11 | The application of our methods to a 32-bit unsigned integer multiplication demonstrated a 90\% circuit depth reduction compared to carry-ripple approaches. | result 1.00 | 0.94 | 0.44 | asserted 1.00 |
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The paper
Original abstract
One of the major promises of quantum computing is the realization of SIMD (single instruction - multiple data) operations using the phenomenon of superposition. Since the dimension of the state space grows exponentially with the number of qubits, we can easily reach situations where we pay less than a single quantum gate per data point for data-processing instructions which would be rather expensive in classical computing. Formulating such instructions in terms of quantum gates, however, still remains a challenging task. Laying out the foundational functions for more advanced data-processing is therefore a subject of paramount importance for advancing the realm of quantum computing. In this paper, we introduce the formalism of encoding so called-semi-boolean polynomials. As it turns out, arithmetic $\mathbb{Z}/2^n\mathbb{Z}$ ring operations can be formulated as semi-boolean polynomial evaluations, which allows convenient generation of unsigned integer arithmetic quantum circuits. For arithmetic evaluations, the resulting algorithm has been known as Fourier-arithmetic. We extend this type of algorithm with additional features, such as ancilla-free in-place multiplication and integer coefficient polynomial evaluation. Furthermore, we introduce a tailor-made method for encoding signed integers succeeded by an encoding for arbitrary floating-point numbers. This representation of floating-point numbers and their processing can be applied to any quantum algorithm that performs unsigned modular integer arithmetic. We discuss some further performance enhancements of the semi boolean polynomial encoder and finally supply a complexity estimation. The application of our methods to a 32-bit unsigned integer multiplication demonstrated a 90\% circuit depth reduction compared to carry-ripple approaches.
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How the literature uses it
5 arXiv papers cite this in 5 sentences
Papers citing it · 1 sentence without a confident label
Uses its method or tool 3 papers
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We use the ones introduced in Ref. [ref] , which perform the following operations
qLUE: A Quantum Clustering Algorithm for Multi- Dimensional Datasets · Grover's Algorithm · 1.00
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We will use quantum arithmetic gates [ref] specifically an addition gate, $$, and a multiplication gate, $$.
Quantum mean estimation for lattice field theory · Pi with and without a sign problem · 0.99
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Projections can be generally performed by unitary quantum gates, while the multiplication by the matrix elements of $g_αβ$ could be implemented, e.g., by quantum floating-point arithmetic [ref] .
Combinatorial summation of Feynman diagrams: Equation of state of the 2D SU(N) Hubbard model · Vector variant and quantum speed-up · 0.83
Cites it as background 1 paper
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Other numerical representations may be used instead, with the usual trade-offs between qubit count, arithmetic depth, and numerical precision; see Refs. [ref] .
A Quantum Collocation Approach to One-Dimensional Boundary Value Problems with Coherent Amplitude Amplification · Quantum Registers and Encodings · 0.95
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