Computing prime factors with a Josephson phase qubit quantum processor

Journal name:
Nature Physics
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Published online

A quantum processor can be used to exploit quantum mechanics to find the prime factors of composite numbers1. Compiled versions of Shor’s algorithm and Gauss sum factorizations have been demonstrated on ensemble quantum systems2, photonic systems3, 4, 5, 6 and trapped ions7. Although proposed8, these algorithms have yet to be shown using solid-state quantum bits. Using a number of recent qubit control and hardware advances9, 10, 11, 12, 13, 14, 15, 16, here we demonstrate a nine-quantum-element solid-state quantum processor and show three experiments to highlight its capabilities. We begin by characterizing the device with spectroscopy. Next, we produce coherent interactions between five qubits and verify bi- and tripartite entanglement through quantum state tomography10, 14, 17, 18. 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.

At a glance


  1. Architecture and operation of the quantum processor.
    Figure 1: Architecture and operation of the quantum processor.

    a, Photomicrograph of the sample, fabricated with aluminium (coloured) on sapphire substrate (dark). b, Schematic of the quantum processor. Each phase qubit Qi is capacitively coupled to the central half-wavelength bus resonator B and a quarter-wavelength memory resonator Mi. The control lines carry gigahertz microwave pulses to produce single-qubit operations. Each Qi is coupled to a superconducting quantum interference device (SQUID) for single-shot readout. c, Illustration of quantum processor operation. By applying pulses on each control line, each qubit frequency is tuned in and out of resonance with B (M) to perform entangling (memory) operations. d, Swap spectroscopy16 for all four qubits. Qubit excited state |eright fence probability Pe (colour scale) versus frequency (vertical axis) and interaction time Δτ. The centres of the chevron patterns gives the frequencies of the resonators B, M1–M4, f=6.1,6.8,7.2,7.1,6.9GHz, respectively. The oscillation periods give the coupling strengths between Qi and B (Mi), which are all ( ).

  2. Rapid entanglement for two to four qubits.
    Figure 2: Rapid entanglement for two to four qubits.

    ac, The measured state occupation probabilities PQ1−4 (colour) and Pb (black) for increasing number of participating qubits N={2,3,4} versus interaction time Δτ. In all cases, B is first prepared in the n=1 Fock state10 and the participating qubits are then tuned on resonance with B for the interaction time Δτ. The single excitation begins in B, spreads to the participating qubits and then returns to B. These coherent oscillations continue for a time Δτ and increase in frequency with each additional qubit. d, Oscillation frequency of Pb for increasing numbers of participating qubits. The error bars indicate the −3dB point of the Fourier transformed Pb data. The inset schematics illustrate which qubits participate. The coupling strength increases as , plotted as a black line fit to the data, with . e,f, The real part of the reconstructed density matrices from QST. e, Bell singlet with fidelity FBell=left fenceψs|ρBell|ψsright fence=0.89±0.01 and EOF=0.70. f, Three-qubit with fidelity FW=left fenceW|ρW|Wright fence=0.69±0.01. The measured imaginary parts (not shown) are found to be small, with (e) and (f), as expected theoretically.

  3. Compiled version of Shor/'s algorithm.
    Figure 3: Compiled version of Shor’s algorithm.

    a, Four-qubit circuit to factor , with co-prime a=4. The three steps in the algorithm are initialization (Init), modular exponentiation and the quantum Fourier transform, which computes and returns the period r=2. b, Recompiled three-qubit version of Shor’s algorithm. The redundant qubit Q1 is removed by noting that HH=I. Circuits a and b are equivalent for this specific case. The three steps of the runtime analysis are labelled 1,2 and 3. c, CNOT gates are realized using an equivalent CZ circuit. d, Step 1: Bell singlet between Q2 and Q3 with fidelity FBell=left fenceψs|ρBell|ψsright fence=0.75±0.01 and EOF=0.43. e, Step 2: Three-qubit between Q2, Q3 and Q4 with fidelity FGHZ=left fenceGHZ|ρGHZ|GHZright fence=0.59±0.01. f, Step 3: QST after running the complete algorithm. The three-qubit |GHZright fence is rotated into |ψ3right fence=H2|GHZright fence=(|gggright fence+|eggright fence+|geeright fence−|eeeright fence)/2 with fidelity F=0.55. g,h, The density matrix of the single-qubit output register Q2 formed by: tracing-out Q3 and Q4 from f (g), and directly measuring Q2 with QST (h), both with and Sl=0.78. From 1.5×105 direct measurements, the output register returns the period r=2, with probability 0.483±0.003, yielding the prime factors 3 and 5. i, The density matrix of the single-qubit output register without entangling gates, H2H2|gright fence=I|gright fence. Ideally this calibration algorithm returns r=0 100% of the time. Compared with the single quantum state |ψoutright fence=|gright fence, the fidelity Fcal=left fenceψg|ρcal|ψgright fence=0.83±0.01, which is less than unity owing to the energy relaxation.


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Author information


  1. Department of Physics, University of California, Santa Barbara, California 93106, USA

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


E.L. fabricated the sample, performed the experiments and analysed the data. E.L. and J.M.M. designed the custom electronics. E.L., M.M. and D.S. contributed to software infrastructure. All authors contributed to the fabrication process, qubit design, experimental set-up and manuscript preparation.

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