Computing prime factors with a Josephson phase qubit quantum processor

Lucero, Erik; Barends, R.; Chen, Yinwei; Kelly, J.; Mariantoni, M.; Megrant, A.; O’Malley, P.; Sank, D.; Vainsencher, A.; Wenner, J.; White, T.; Yin, Yi; Cleland, A. N.; Martinis, John M.

doi:10.1038/nphys2385

Cited by 289 publications

(282 citation statements)

References 31 publications

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“…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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Section: Introductionmentioning

confidence: 99%

“…After m − n repetitions we arrive at equation (6), and the problem is reduced to order finding with the root, a.…”

Section: Factoring With Odd Ordersmentioning

confidence: 99%

Odd orders in Shor’s factoring algorithm

Lawson

2015

Quantum Inf Process

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“…Superconducting qubits are a promising candidate for the realization of a quantum computer [1][2][3][4][5], owing in large part to the success of circuit QED (CQED), where those qubits are coupled to microwave resonators [6][7][8]. There is a multitude of designs of such qubits [2].…”

Section: Introductionmentioning

confidence: 99%

Single-qubit gates in frequency-crowded transmon systems

et al. 2013

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Recent experimental work on superconducting transmon qubits in three-dimensional (3D) cavities shows that their coherence times are increased by an order of magnitude compared to their two-dimensional cavity counterparts. However, to take advantage of these coherence times while scaling up the number of qubits it is advantageous to address individual qubits which are all coupled to the same 3D cavity fields. The challenge in controlling this system comes from spectral crowding, where the leakage transition of qubits is close to computational transitions in other qubits. Here, it is shown that fast pulses are possible which address single qubits using two-quadrature control of the pulse envelope, while the derivative removal by adiabatic gate method of Motzoi et al. [Phys. Rev. Lett. 103, 110501 (2009)] alone only gives marginal improvements over the conventional Gaussian pulse shape. On the other hand, a first-order result using the Magnus expansion gives a fast analytical pulse shape which gives a high-fidelity gate for a specific gate time, up to a phase factor on the second qubit. Further numerical analysis corroborates these results and yields to even faster gates, showing that leakage-state anharmonicity does not provide a fundamental quantum speed limit.

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“…1(b)] [23] using up to three superconducting qubits, among which the initial logical qubit state jΨi ¼ j0i − ij1i (here and below we neglect the normalization constant) is relayed and its phase information is probed after the total relay time τ. We use two types of superconducting circuits in which dephasing noises differ very much in magnitude: one features three phase qubits, each capacitively coupled to a common resonator [24,25] [see Fig. 1(a)], and the other one features two Xmon qubits with much reduced dephasing noises, each coupled capacitively as well to a common resonator.…”

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confidence: 99%

Suppression of Dephasing by Qubit Motion in Superconducting Circuits

Averin

Zhong

et al. 2016

Phys. Rev. Lett.

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We suggest and demonstrate a protocol which suppresses the low-frequency dephasing by qubit motion, i.e., transfer of the logical qubit of information in a system of n ≥ 2 physical qubits. The protocol requires only the nearest-neighbor coupling and is applicable to different qubit structures. Our analysis of its effectiveness against noises with arbitrary correlations, together with experiments using up to three superconducting qubits, shows that for the realistic uncorrelated noises, qubit motion increases the dephasing time of the logical qubit as ffiffiffi n p . In general, the protocol provides a diagnostic tool for measurements of the noise correlations. DOI: 10.1103/PhysRevLett.116.010501 Development of superconducting qubits [1][2][3][4][5][6][7] has reached the stage where it is interesting to discuss possible architectures of the quantum information processing circuits. The common feature of any quantum computation process of even moderate complexity is the requirement of information transfer between different elements of the qubit circuit. The most straightforward way of achieving this transfer is to physically move the quantum states representing the qubits of information along the circuit. In the case of superconducting qubits, potential for such a direct motion of logical qubits is offered by so-called negative-inductance SQUIDs [8,9], but operation of these circuits in the quantum regime [10] still needs to be demonstrated experimentally. Another method of transferring logical qubits between different physical qubits, already developed in experiments and adopted in this work, is based on creating controlled qubit-qubit interaction through coupling to a common resonator bus [5,[11][12][13]] (see Fig. 1). The goal of this Letter is to demonstrate that, in addition to its main function, the transfer of information between different circuit elements designed to perform different functions has an additional notable benefit: suppression of the low-frequency dephasing. We also show that it can be used to measure the noise correlations and, in this way, diagnose the primary sources of the noises.The basic mechanism of the noise suppression by qubit motion relies on the fact that the low-frequency noise is typically produced by fluctuators-see, e.g., Refs. [14,15] -in the form of impurity charges or magnetic moments, localized in each individual physical qubit, and, therefore, is not correlated among them. Motion of a logical qubit between different physical qubits limits the correlation time of the effective noise seen by this qubit and, therefore, suppresses its decoherence rate. This effect is qualitatively similar to the motional narrowing of the NMR lines [16], with the main difference that it is based on the controlled transfer of the qubit state, not random thermal motion as in NMR. Our protocol also has some similarities to the dynamic decoupling schemes-see, e.g., Refs. [17,18]where the qubit-noise interaction is suppressed by changing the qubit state in the effectively constant noise, while th...

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Computing prime factors with a Josephson phase qubit quantum processor

Cited by 289 publications

References 31 publications

Odd orders in Shor’s factoring algorithm

Odd orders in Shor’s factoring algorithm

Single-qubit gates in frequency-crowded transmon systems

Suppression of Dephasing by Qubit Motion in Superconducting Circuits

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