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arXiv:1611.07995 · Quantum Information and Computation

Factoring using 2n+2 qubits with Toffoli based modular multiplication

Thomas Häner, Martin Roetteler, Krysta M. Svore

Published May 1, 2017• 90 citations
arXivHTMLPDF

Abstract

We describe an implementation of Shor’s quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The circuit depth and the overall gate count are in O(n 3 ) and O(n 3 log n), respectively. We thus achieve the same space and time costs as Takahashi et al. [1], while using a purely classical modular multiplication circuit. As a consequence, our approach evades most of the cost overheads originating from rotation synthesis and enables testing and localization of some faults in both, the logical level circuit and an actual quantum hardware implementation. Our new (in-place) constant-adder, which is used to construct the modular multiplication circuit, uses only dirty ancilla qubits and features a circuit size and depth in O(n log n) and O(n), respectively.

DOI
10.26421/qic17.7-8-7

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We describe an implementation of Shor’s quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The circuit depth and the overall gate count are in O(n 3 ) and O(n 3 log n), respectively. We thus achieve the same space and time costs as Takahashi et al. [1], while using a purely classical modular multiplication circuit. As a consequence, our approach evades most of the cost overheads originating from rotation synthesis and enables testing and localization of some faults in both, the logical level circuit and an actual quantum hardware implementation. Our new (in-place) constant-adder, which is used to construct the modular multiplication circuit, uses only dirty ancilla qubits and features a circuit size and depth in O(n log n) and O(n), respectively.
We try to minimize the number of qubits needed to factor an integer of n bits using Shor's algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n^3 lg(n)) elementary quantum gates in a depth of O(n^3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored. Keywords: Factorization, quantum circuits, modular arithmetics
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Reversible circuits for modular multiplication Cx%MCx\%MCx%M with x<Mx<Mx<M arise as components of modular exponentiation in Shor's quantum number-factoring algorithm. However, existing generic constructions focus on asymptotic gate count and circuit depth rather than actual values, producing fairly large circuits not optimized for specific CCC and MMM values. In this work, we develop such optimizations in a bottom-up fashion, starting with most convenient CCC values. When zero-initialized ancilla registers are available, we reduce the search for compact circuits to a shortest-path problem. Some of our modular-multiplication circuits are asymptotically smaller than previous constructions, but worst-case bounds and average sizes remain Θ(n2)\Theta(n^2)Θ(n2). In the context of modular exponentiation, we offer several constant-factor improvements, as well as an improvement by a constant additive term that is significant for few-qubit circuits arising in ongoing laboratory experiments with Shor's algorithm.
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qu trits . Past work with qutrits has demonstrated only constant factor improvements, owing to the log 2 (3) binary-to-ternary compression factor. We present a novel technique, intermediate qutrits, to achieve sublinear depth decompositions of the Generalized Toffoli and other arithmetic circuits using no additional ancilla—a significant improvement over linear depth for the best qubit-only equivalents. For example, our Generalized Toffoli construction features a 70× improvement in two-qudit gate count over a qubit-only decomposition. This results in circuit cost reductions for important algorithms like quantum neurons, Grover search, and even Shor’s algorithm. Using a previously developed simulator with near-term noise models, we demonstrate for these models over 90% mean reliability (fidelity) for the Toffoli construction, versus under 30% for the qubit-only baseline. For our other constructions, such as the Incrementer, the A + B adder and the +K adder, we demonstrate the power of intermediate qutrits in producing asymptotic depth improvements with no additional ancilla. Together, these results suggest qutrits offer a promising path toward scaling quantum computation.
We present an arithmetic circuit performing constant modular addition having O(n)O(n)O(n) depth of Toffoli gates and using a total of n+3n+3n+3 qubits. This is an improvement by a factor of two compared to the width of the state-of-the-art Toffoli-based constant modular adder. The advantage of our adder, compared to the ones operating in the Fourier basis, is that it does not require small-angle rotations and their Clifford+T\text{Clifford}+TClifford+T decomposition. Our circuit uses a recursive adder combined with the modular addition scheme proposed by Vedral et al. The circuit is implemented and verified exhaustively with quantify, an open-sourced framework. We also report on the Clifford+T\text{Clifford}+TClifford+T cost of the circuit.
We present a novel and efficient, in terms of circuit depth, design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the Quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a quantum modular multiplier which is the fundamental block for the modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about 2000n22000n^22000n2 and requires 9n+29n+29n+2 qubits, where nnn is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is 1600n31600n^31600n3. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.