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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 Computing prime factors with a Josephson phase qubit quantum processor Erik Lucero,1 R. Barends,1 Y. Chen,1 J. Kelly,1 M. Mariantoni,1 A. Megrant,1 P. O'Malley,1 D. Sank,1 A. Vainsencher,1 J. Wenner,1 T. White,1 Y. Yin,1 A. N. Cleland,1 and John M. Martinis1 1Department of Physics, University of California, Santa Barbara, CA 93106, USA 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 ensem- ble quantum systems[2] and photonic systems[3{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{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 to- mography (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 pro- vide the resources necessary to factor larger composite numbers and run more intricate quantum algorithms. In this experiment, we scaled-up from an architec- this algorithm involves the challenge of combining both ture initially implemented with two qubits and three res- single- and coupled-qubit gates in a meaningful sequence. onators [7] to a nine-element quantum processor (QuP) We constructed the full factoring sequence by first per- capable of realizing rapid entanglement and a compiled forming automatic calibration of the individual gates and version of Shor's algorithm. The device is composed then combined them, without additional tuning, so as to of four phase qubits and five superconducting coplanar factor the composite number N = 15 with co-prime a = 4, waveguide (CPW) resonators, where the resonators are (where 1 < a < N and the greatest common divisor be- used as qubits by accessing only the two lowest levels. tween a and N is 1). We also checked for entanglement Four of the five CPWs can be used as quantum mem- at three points in the algorithm using QST. ory elements as in Ref. [7] and the fifth can be used to Figure 1a shows a micrograph of the QuP, made on mediate entangling operations. a sapphire substrate using Al/AlOx/Al Josephson junc- The QuP can create entanglement and execute quan- tions. Figure 1b shows a complete schematic of the de- tum circuits[16, 17] with high-fidelity single-qubit gates vice. Each qubit Qi is individually controlled using a bias (X, Y , Z, and H), [18, 19]combined with swaps and coil that carries dc, rf- and GHz-pulses to adjust the qubit controlled-phase (Cφ) gates[7, 13, 20], where one qubit in- frequency and to pulse microwaves for manipulating and teracts with a resonator at a time. The QuP can also uti- measuring the qubit state. Each qubit's frequency can be lize \fast-entangling logic" by bringing all participating adjusted over an operating range of ∼ 2 GHz, allowing us qubits on resonance with the resonator at the same time to couple each qubit to the other quantum elements on to generate simultaneous entanglement[21]. At present, the chip. Each Qi is connected to a memory resonator this combination of entangling capabilities has not been Mi, as well as the bus B, via interdigitated capacitors. demonstrated on a single device. Previous examples have Although the coupling capacitors are fixed, Fig. 1c illus- shown: spectroscopic evidence of the increased coupling trates how the effective interaction can be controlled by arXiv:1202.5707v1 [quant-ph] 26 Feb 2012 for up to three qubits coupled to a resonator[14], as well tuning the qubits into or near resonance with the cou- as coherent interactions between two and three qubits pling bus (coupling \on") or detuning Qi to fB ±500 MHz with a resonator[12], although these lacked tomographic (coupling “off")[24]. evidence of entanglement. The QuP is mounted in a superconducting aluminum Here we show coherent interactions for up to four sample holder and cooled in a dilution refrigerator to qubits with a resonator and verify genuine bi- and tripar- ∼ 25 mK. Qubit operation and calibration are similar tite entanglement including Bell [9] and jWi-states [10] to previous works[7{10, 13],with the addition of an au- with quantum state tomography (QST). This QuP has tomated calibration process[25]. As shown in Fig.1d, we the further advantage of creating entanglement at a rate used swap spectroscopy[7] to calibrate all nine of the en- more than twice that of previous demonstrations[10, 12]. gineered quantum elements on the QuP, the four phase In addition to these characterizations, we chose to im- qubits (Q1 − Q4), the four quarter-wave CPW quantum plement a compiled version of Shor's algorithm[22, 23], memory resonators (M1 − M4), and one half-wave CPW in part for its historical relevance[16] and in part because bus resonator (B). The coupling strengths between Qi 2 a and B (Mi) were measured to be within 5 % (10 %) of the design values. M The qubit-resonator interaction can be described by Q2 2 the Jaynes-Cummings model Hamiltonian[26] Hint = M − + Q 4 P y Control 1 i(¯hgi=2)(a σi +aσi ), where gi is the coupling strength y B between the bus resonator B and the qubit Qi, a and M SQUID 1 Q4 a are respectively the photon creation and annihila- + − M3 tion operators for the resonator, σi and σi are respec- Q 1mm 3 tively the qubit Qi raising and lowering operators, and ¯h = h=2π. The dynamics during the interaction between the i = f1; 2; 3; 4g qubits and the bus resonator are shown in Fig.1c, and Fig.2 a, b, c respectively. b For these interactions the qubits Q1 −Q4 are initialized Q2 M2 M4 Q4 in the ground state jggggi and tuned off-resonance from λ/2 B at an idle frequency f ∼ 6:6 GHz. Q1 is prepared in dc, rf, GHz B the excited state jei via a π-pulse. B is then pumped Control Q1 M1 M3 Q3 into the first Fock state n = 1 by tuning Q1 on resonance SQUID λ/4 (f ∼ 6:1 GHz) for a duration 1=2g1 = τ ∼ 9 ns, long c enough to complete an iSWAP operation between Q1 and M2 M B, j0i ⊗ jegggi ! j1i ⊗ jggggi [8]. 3 M4 M1 Q3 The participating qubits are then tuned on resonance Q 1 Q (f ∼ 6:1 GHz) and left to interact with B for an interac- rf 2 B Q4 tion time ∆τ. Figures 2 a,b,c show the probability PQi of measuring the participating qubits in the excited state, frequency and the probability PB of B being in the n = 1 Fock d Q1 Q2 Q3 Q4 state, versus ∆τ. At the beginning of the interaction the M3 7.2 excitation is initially concentrated in B (P maximum) M4 B M then spreads evenly between the participating qubits (P 1 M2 6.9 B 6.8 minimum) and finally returns back to B, continuing as a coherent oscillation during this interaction time. When the qubits are simultaneously tuned on reso- B nance with B they interact with an effective coupling 6.1 strengthp g ¯N that scales with the number N of qubits frequency [GHz] as N[14], analogous to a single qubit coupled to a 0 200 0 200 0 200 0 200 resonatorp in a n-photon Fock state[8]. For N qubits, Δτ (ns) Δτ (ns) Δτ (ns) Δτ (ns) P 2 1=2 g¯N = Ng¯, whereg ¯ = [1=N( i=1;N gi )] . The 0.0 1.0 Pe oscillation frequency of PB for each of the four cases i = f1; 2; 3; 4g is shown in Fig.2 d. These results are FIG. 1: Architecture and operation of the quantum pro- similar to Ref.[14], but with a larger number N of qubitsp cessor (QuP). a, Photomicrograph of the sample, fabricated interacting with the resonator, we can confirm the N with aluminum (colored) on sapphire substrate (dark). b, scaling of the coupling strength with N. From these data Schematic of the QuP. Each phase qubit Qi is capacitively we find a mean value ofg ¯ = 56:5 ± 0:05 MHz. coupled to the central half-wavelength bus resonator B and a By tuning the qubits on resonance for a specific inter- quarter-wavelength memory resonator Mi. The control lines carry GHz microwave pulses to produce single qubit oper- action time τ, corresponding to the first minimum of PB in Fig. 2a, b, we can generate Bell singlets j i = (jgei − ations. Each Qi is coupled to a superconducting quantum p S p interference device (SQUID) for single-shot readout. c, Illus- jegi)= 2 and jWi states jWi = (jggei+jgegi+jeggi)= 3. tration of QuP operation. By applying pulses on each control Stopping the interaction at this time (τBell = 6:5 ns line, each qubit frequency is tuned in and out of resonance and τW = 5:1 ns) leaves the single excitation evenly dis- with B (M) to perform entangling (memory) operations.
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