Demonstration of a Compiled Version of Shor’s Quantum Factoring Algorithm Using Photonic Qubits

Lu, Chao‐Yang; Browne, Dan E.; Yang, Tao; Pan, Jian-Wei

doi:10.1103/physrevlett.99.250504

Cited by 239 publications

(190 citation statements)

References 41 publications

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“…Despite the order being odd, the algorithm successfully returns the factors, 3 = gcd(4 3 2 + 1, 21) and 7 = gcd(4 3 2 − 1, 21), which are integer because the coprime is square.…”

Section: Factoring With Odd Ordersmentioning

confidence: 99%

“…Take factoring 21 with coprime 16. The order three leads to the trivial factors gcd(16 3 2 + 1, 21) = 1 and gcd(16 3 2 − 1, 21) = 21 because the condition (2) is not met, 21|16 3 2 − 1. So how often are the factors found from odd orders?…”

Section: Factoring With Odd Ordersmentioning

confidence: 99%

“…SFA has been extensively studied theoretically, but it has not yet been convincingly demonstrated in the lab; the difficulty of controlling quantum systems means just a handful of experiments have been done, to test the basic principles [1][2][3][4][5][6]. These experiments are too simple to be of practical use but are, nonetheless, important.…”

Section: Introductionmentioning

confidence: 99%

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Odd orders in Shor’s factoring algorithm

Lawson

2015

Quantum Inf Process

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Shor's factoring algorithm (SFA) finds the prime factors of a number, N = p1p2, exponentially faster than the best known classical algorithm. Responsible for the speed-up is a subroutine called the quantum order finding algorithm (QOFA) which calculates the order -the smallest integer, r, satisfying a r mod N = 1, where a is a randomly chosen integer coprime to N (meaning their greatest common divisor is one, gcd(a, N ) = 1). Given r, and with probability not less than 1/2, the factors are given by p1 = gcd(a r 2 − 1, N ) and p2 = gcd(a r 2 + 1, N ). For odd r it is assumed the factors cannot be found (since a r 2 is not generally integer) and the QOFA is relaunched with a different value of a. But a recent paper [E. Martin-Lopez et al.: Nat Photon 6, 773 (2012)] noted that the factors can sometimes be found from odd orders if the coprime is square.This raises the question of improving SFA's success probability by considering odd orders. We show that an improvement is possible, though it is small. We present two techniques for retrieving the order from apparently useless runs of the QOFA: not discarding odd orders; and looking out for new order finding relations in the case of failure. In terms of efficiency, using our techniques is equivalent to avoiding square coprimes and disregarding odd orders, which is simpler in practice. Even still, our techniques may be useful in the near future, while demonstrations are restricted to factoring small numbers. The most convincing demonstrations of the QOFA are those that return a non-power-of-two order, making odd orders that lead to the factors attractive to experimentalists.

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“…Despite the order being odd, the algorithm successfully returns the factors, 3 = gcd(4 3 2 + 1, 21) and 7 = gcd(4 3 2 − 1, 21), which are integer because the coprime is square.…”

Section: Factoring With Odd Ordersmentioning

confidence: 99%

Section: Factoring With Odd Ordersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Odd orders in Shor’s factoring algorithm

Lawson

2015

Quantum Inf Process

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“…Compiled versions of Shor's algorithm have been demonstrated on ensemble quantum systems [2] and photonic systems [3][4][5], however this has yet to be shown using solid state quantum bits (qubits). Two advantages of superconducting qubit architectures are the use of conventional microfabrication techniques, which allow straightforward scaling to large numbers of qubits, and a toolkit of circuit elements that can be used to engineer a variety of qubit types and interactions [6,7].…”

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confidence: 99%

Computing prime factors with a Josephson phase qubit quantum processor

et al. 2012

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A quantum processor (QuP) can be used to exploit quantum mechanics to find the prime factors of composite numbers [1]. Compiled versions of Shor's algorithm have been demonstrated on ensemble quantum systems [2] and photonic systems [3][4][5], however this has yet to be shown using solid state quantum bits (qubits). Two advantages of superconducting qubit architectures are the use of conventional microfabrication techniques, which allow straightforward scaling to large numbers of qubits, and a toolkit of circuit elements that can be used to engineer a variety of qubit types and interactions [6,7]. Using a number of recent qubit control and hardware advances [7][8][9][10][11][12][13], here we demonstrate a nine-quantum-element solid-state QuP and show three experiments to highlight its capabilities. We begin by characterizing the device with spectroscopy. Next, we produces coherent interactions between five qubits and verify bi-and tripartite entanglement via quantum state tomography (QST) [8,12,14,15]. In the final experiment, we run a three-qubit compiled version of Shor's algorithm to factor the number 15, and successfully find the prime factors 48 % of the time. Improvements in the superconducting qubit coherence times and more complex circuits should provide the resources necessary to factor larger composite numbers and run more intricate quantum algorithms.In this experiment, we scaled-up from an architecture initially implemented with two qubits and three resonators [7] to a nine-element quantum processor (QuP) capable of realizing rapid entanglement and a compiled version of Shor's algorithm. The device is composed of four phase qubits and five superconducting coplanar waveguide (CPW) resonators, where the resonators are used as qubits by accessing only the two lowest levels. Four of the five CPWs can be used as quantum memory elements as in Ref. [7] and the fifth can be used to mediate entangling operations.The QuP can create entanglement and execute quantum circuits [16,17] with high-fidelity single-qubit gates (X, Y , Z, and H), [18,19]combined with swaps and controlled-phase (C φ ) gates [7,13,20], where one qubit interacts with a resonator at a time. The QuP can also utilize "fast-entangling logic" by bringing all participating qubits on resonance with the resonator at the same time to generate simultaneous entanglement [21]. At present, this combination of entangling capabilities has not been demonstrated on a single device. Previous examples have shown: spectroscopic evidence of the increased coupling for up to three qubits coupled to a resonator [14], as well as coherent interactions between two and three qubits with a resonator [12], although these lacked tomographic evidence of entanglement.Here we show coherent interactions for up to four qubits with a resonator and verify genuine bi-and tripartite entanglement including Bell [9] and |W -states [10] with quantum state tomography (QST). This QuP has the further advantage of creating entanglement at a rate more than twice that of previous demonst...

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“…However, in the special case where the order is a power of two and r = 2 p , the peaks correspond exactly to the logical states of p qubits 19 such that the output is equivalent to that of an incoherent mixture of p qubits (any additional control qubits remain unentangled throughout the algorithm and simply encode the trailing zeros in the uniform distribution). Factoring N = 15 gives either order 2 or order 4 for each of its co-primes 6-9 ; independent verification of entanglement is therefore required 7,8 . For this reason we focus on N = 21 with the co-prime x = 4 to give order 20 r = 3.…”

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confidence: 99%

Experimental realization of Shor's quantum factoring algorithm using qubit recycling

et al. 2012

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Quantum computational algorithms exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms 1-3 . Shor's quantum algorithm 4 for fast number factoring is a key example and the prime motivator in the international effort to realise a quantum computer 5 . However, due to the substantial resource requirement, to date, there have been only four small-scale demonstrations 6-9 . Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which the n qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol [10][11][12] . Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N = 21. The algorithmic output is distinguishable from noise, in contrast to previous demonstrations. These results point to larger-scale implementations of Shor's algorithm by harnessing scalable resource reductions applicable to all physical architectures.Shor's factoring algorithm consists of a quantum order finding algorithm, preceded and succeeded by various classical routines. While the classical tasks are known to be efficient on a classical computer, order finding is understood to be intractable classically. However, it is known that this part of the algorithm can be performed efficiently on a quantum computer. To determine the prime factors of an odd integer N , one chooses a coprime of N , x. The order r relates x to N according to x r mod N = 1, and can be used to obtain the factors, given by the greatest common divisor gcd(x r 2 ± 1, N ). The quantum order finding circuit involves two registers: a work register and a control register. In the standard protocol, the work register performs modular arithmetic with m = log 2 N qubits, enough to encode the number N , and the n qubit control register provides the algorithmic output, with n bits of precision.Measuring the control register in the computational basis will yield a result of k2 n /r where k is an integer between 0 and r − 1, with the value of k occurring probabilistically. Dividing the result by 2 n gives the first n bits of k/r and r may be found with classical processing, using the continued fraction algorithm. For large n, and a perfectly functioning circuit, the output probability distribution of the control register is a series of well defined peaks at values of k2 n /r (Fig. 1b). (See Appendix for details.)Here we implement an iterative version of the order finding algorithm 10,11 , in which the control register contains only a single qubit which is recycled n times, using a sequence of measurement and feed-forward operations, with each step providing an additional bit of precision (Fig. 1a). Reducing the number of qubits in quantum simulations and quantum chemistry has been achieved with recursive phase estimation 13-16 , while ground state projections have been demonstrated by exploiting similar techniques in NMR 17 .The iterative version of...

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Demonstration of a Compiled Version of Shor’s Quantum Factoring Algorithm Using Photonic Qubits

Cited by 239 publications

References 41 publications

Odd orders in Shor’s factoring algorithm

Odd orders in Shor’s factoring algorithm

Computing prime factors with a Josephson phase qubit quantum processor

Experimental realization of Shor's quantum factoring algorithm using qubit recycling

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