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Computing Prime Factors with a Josephson Phase Qubit Quantum Processor

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Computing Prime Factors with a Josephson Phase Qubit Quantum Processor LETTERS PUBLISHED ONLINE: 19 AUGUST 2012 | DOI: 10.1038/NPHYS2385 Computing prime factors with a Josephson phase qubit quantum processor 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 and John M. Martinis* A quantum processor can be used to exploit quantum mechan- In addition to these characterizations, we chose to implement ics to find the prime factors of composite numbers1. Compiled a compiled version of Shor's algorithm25,26, in part for its historical versions of Shor’s algorithm and Gauss sum factorizations relevance19 and in part because this algorithm involves the challenge have been demonstrated on ensemble quantum systems2, pho- of combining both single- and coupled-qubit gates in a meaningful tonic systems3–6 and trapped ions7. Although proposed8, these sequence. We constructed the full factoring sequence by first algorithms have yet to be shown using solid-state quantum performing automatic calibration of the individual gates and bits. Using a number of recent qubit control and hardware then combined them, without additional tuning, so as to factor advances9–16, here we demonstrate a nine-quantum-element the composite number N D 15 with co-prime a D 4, (where solid-state quantum processor and show three experiments to 1 < a < N and the greatest common divisor between a and N highlight its capabilities. We begin by characterizing the device is 1). We also checked for entanglement at three points in the with spectroscopy. Next, we produce coherent interactions algorithm using QST. between five qubits and verify bi- and tripartite entanglement Figure 1a shows a micrograph of the quantum processor, made 10,14,17,18 through quantum state tomography . In the final experi- on a sapphire substrate using Al=AlOx=Al Josephson junctions. ment, we run a three-qubit compiled version of Shor’s algorithm Figure 1b shows a complete schematic of the device. Each qubit to factor the number 15, and successfully find the prime factors Qi is individually controlled using a bias coil that carries d.c., 48% of the time. Improvements in the superconducting qubit radiofrequency and gigahertz pulses to adjust the qubit frequency coherence times and more complex circuits should provide the and to pulse microwaves for manipulating and measuring the qubit resources necessary to factor larger composite numbers and state. Each qubit's frequency can be adjusted over an operating run more intricate quantum algorithms. range of ∼2 GHz, allowing us to couple each qubit to the other In this experiment, we scaled up from an architecture initially quantum elements on the chip. Each Qi is connected to a memory 16 implemented with two qubits and three resonators to a nine- resonator Mi, as well as the bus B, through interdigitated capacitors. element quantum processor capable of realizing rapid entanglement Although the coupling capacitors are fixed, Fig. 1c illustrates how and a compiled version of Shor's algorithm. The device is composed the effective interaction can be controlled by tuning the qubits into of four phase qubits and five superconducting co-planar waveguide or near resonance with the coupling bus (coupling on) or detuning 27 (CPW) resonators, where the resonators are used as qubits by Qi to fB 500 MHz (coupling off) . accessing only the two lowest levels. Four of the five CPWs can be The quantum processor is mounted in a superconducting used as quantum memory elements as in ref. 16 and the fifth can be aluminium sample holder and cooled in a dilution refrigerator to used to mediate entangling operations. ∼25 mK. Qubit operation and calibration are similar to previous The quantum processor can create entanglement and execute works10–12,15,16, with the addition of an automated calibration quantum algorithms19,20 with high-fidelity single-qubit gates21,22 process28. As shown in Fig. 1d, we used swap spectroscopy16 to (X, Y , Z and H) combined with swap and controlled-phase (Cφ ) calibrate all nine of the engineered quantum elements on the 15,16,23 gates , where one qubit interacts with a resonator at a time. The quantum processor, the four phase qubits (Q1–Q4), the four quantum processor can also use fast-entangling logic by bringing quarter-wave CPW quantum-memory resonators (M1–M4) and all participating qubits on resonance with the resonator at the one half-wave CPW bus resonator (B). The coupling strengths 24 same time to generate simultaneous entanglement . At present, this between Qi and B (Mi) were measured to be within 5% (10%) combination of entangling capabilities has not been demonstrated of the design values. on a single device. Previous examples have shown spectroscopic The qubit–resonator interaction can be described by the Jaynes– 29 D P = yσ − C σ C evidence of the increased coupling for up to three qubits coupled Cummings model Hamiltonian Hint i(hgN i 2)(a i a i ), 17 to a resonator , as well as coherent interactions between two and where gi is the coupling strength between the bus resonator B 14 y three qubits with a resonator , although these lacked tomographic and the qubit Qi, a and a are, respectively, the photon creation σ C σ − evidence of entanglement. and annihilation operators for the resonator, i and i are, Here we show coherent interactions for up to four qubits with respectively, the qubit Qi raising and lowering operators and a resonator and verify genuine bi- and tripartite entanglement hN D h=2π. The dynamics during the interaction between the including Bell11 and jW i states12 with quantum state tomography i D f1;2;3;4g qubits and the bus resonator are shown in Fig. 1c and (QST). This quantum processor has the further advantage of Fig. 2a,b,c respectively. creating entanglement at a rate more than twice that of previ- For these interactions the qubits Q1–Q4 are initialized in the ous demonstrations12,14. ground state jggggi and tuned off-resonance from B at an idle Department of Physics, University of California, Santa Barbara, California 93106, USA. *e-mail: [email protected]. NATURE PHYSICS j VOL 8 j OCTOBER 2012 j www.nature.com/naturephysics 719 © 2012 Macmillan Publishers Limited. All rights reserved. LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS2385 a Q 2 M2 M Q 4 Control 1 B UID M1 SQ Q4 M3 Q 1 mm 3 b Q Q 2 M2 M4 4 λ/2 Control B Q1 Q3 M1 M3 SQUID λ/4 c M M2 3 M4 M1 Q3 Q1 Q RF 2 B Q4 Frequency d Q Q Q Q 1 2 3 M3 4 7.2 M4 M M1 2 6.9 6.8 B Frequency (GHz) 6.1 0 200 0 200 0 200 0 200 Δτ (ns) Δ τ (ns) Δ ττ (ns) Δ (ns) 0.0 1.0 Pe Figure 1 j 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 jei probability Pe (colour scale) versus frequency (vertical axis) and interaction time 1τ. The centres of the chevron patterns gives the frequencies of the resonators B, M1–M4, f D 6:1;6:8;7:2;7:1;6:9 GHz, respectively. The oscillation periods give the coupling strengths ∼ ∼ between Qi and B (Mi), which are all D55 MHz (D20 MHz). 720 NATURE PHYSICS j VOL 8 j OCTOBER 2012 j www.nature.com/naturephysics © 2012 Macmillan Publishers Limited. All rights reserved. NATURE PHYSICS DOI: 10.1038/NPHYS2385 LETTERS a 1.0 N = 2 b P de , Q4 P 0.5 | ψ 〉 , s Q1 P Q Q 1 3 1/2 B 0.0 4 0 20 40 60 80 100 0 Q Δτ (ns) Q2 4 ¬1/2 b 1.0 Q N = 3 1 |gg〉 |ee〉 b P B | 〉 | 〉 , ee gg 3 Q4 P , 0.5 N Q2 Q4 Q2 f P | 〉 , Q W Q1 1 P 1/3 B 0.0 2 0 20 40 60 80 100 Q Δτ (ns) 4 c 1.0 Q b N = 4 1 0 P , B | 〉 Q4 | 〉 eee P ggg , 1 Q3 0.5 | 〉 |ggg〉 P eee , Q2 P 50 70 90 110 130 , Q1 Oscillation frequency (MHz) P 0.0 0 20 40 60 80 100 Δτ (ns) Figure 2 j Rapid entanglement for two to four qubits. a–c, The measured state occupation probabilities PQ1−4 (colour) and Pb (black) for increasing number of participating qubits N D f2;3;4g versus interaction time 1τ. In all cases, B is first prepared in the n D 1 Fock state10 and the participating qubits are then tuned on resonance with B for the interaction time 1τ. The single excitation begins in B, spreads to the participating qubits and then returns to B. These coherent oscillations continue for a time 1τ and increase in frequency with each additional qubit. d, Oscillation frequency of Pb for increasing numbers of participating qubits. The error bars indicatep the −3 dB point of the Fourier transformed Pb data.
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