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arXiv:2108.05949

Quantum Rounding

Rajiv Krishnakumar, William J. Zeng

Published Aug 12, 2021• 0 citations
arXivHTMLPDF

Abstract

We introduce new rounding methods to improve the accuracy of finite precision quantum arithmetic. These quantum rounding methods are applicable when multiple samples are being taken from a quantum program. We show how to use multiple samples to stochastically suppress arithmetic error from rounding. We benchmark these methods on the multiplication of fixed-point numbers stored in quantum registers. We show that the gate counts and depths for multiplying to a target accuracy can be reduced by approximately 2-3X over state of the art methods while using roughly the same number of qubits.

DOI
10.48550/arxiv.2108.05949

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We introduce new rounding methods to improve the accuracy of finite precision quantum arithmetic. These quantum rounding methods are applicable when multiple samples are being taken from a quantum program. We show how to use multiple samples to stochastically suppress arithmetic error from rounding. We benchmark these methods on the multiplication of fixed-point numbers stored in quantum registers. We show that the gate counts and depths for multiplying to a target accuracy can be reduced by approximately 2-3X over state of the art methods while using roughly the same number of qubits.
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This is an investigation of the limits of quantum circuit simulation with Schrodinger's formulation and low precision arithmetic. The goal is to estimate how much memory can be saved in simulations that involve random, maximally entangled quantum states. An arithmetic polar representation of BBB bits is defined for each quantum amplitude and a normalization procedure is developed to minimize rounding errors. Then a model is developed to quantify the cumulative errors on a circuit of QQQ qubits and GGG gates. Depending on which regime the circuit operates, the model yields explicit expressions for the maximum number of effective gates that can be simulated before rounding errors dominate the computation. The results are illustrated with random circuits and the quantum Fourier transform.
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.
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Quantum simulators are a ubiquitous part of quantum software development. This is especially true for the current NISQ-era but will probably remain true for a long time as error-free controlled testing is hardly possible on real devices. However, quantum simulators are implemented in classical languages and thus fall prey to rounding errors. We present bounds on these errors based on the number of gates and qubits alone. This can be crucial to rule out rounding errors prior to executing the circuit. Additionally, we show that these theoretical bounds are sufficient to essentially rule out relevant rounding errors for the simulation of small to medium-sized circuits and even for larger circuits under the assumption that only 1- and 2-qubit gates are used.
The multiplicative depth of a logic network over the gate basis {∧,⊕,¬}\{\land, \oplus, \neg\}{∧,⊕,¬} is the largest number of ∧\land∧ gates on any path from a primary input to a primary output in the network. We describe a dynamic programming based logic synthesis algorithm to reduce the multiplicative depth in logic networks. It makes use of cut enumeration, tree balancing, and exclusive sum-of-products (ESOP) representations. Our algorithm has applications to cryptography and quantum computing, as a reduction in the multiplicative depth directly translates to a lower TTT-depth of the corresponding quantum circuit. Our experimental results show improvements in TTT-depth over state-of-the-art methods and over several hand-optimized quantum circuits for instances of AES, SHA, and floating-point arithmetic.