Abstract
Quantum gates must perform reliably when operating on standard input basis states and on complex superpositions thereof. Experiments using superconducting qubits have validated truth tables for particular implementations of, for example, the controlledNOT gate^{1,2}, but have not fully characterized gate operation for arbitrary superpositions of input states. Here we demonstrate the use of quantum process tomography^{3,4} (QPT) to fully characterize the performance of a universal entangling gate between two superconducting qubits. Process tomography permits complete gate analysis, but requires precise preparation of arbitrary input states, control over the subsequent qubit interaction and ideally simultaneous singleshot measurement of output states. In recent work, it has been proposed to use QPT to probe noise properties^{5} and time dynamics^{6} of qubit systems and to apply techniques from control theory to create scalable qubit benchmarking protocols^{7,8}. We use QPT to measure the fidelity and noise properties^{5} of an entangling gate. In addition to demonstrating a promising fidelity, our entangling gate has an ontooff ratio of 300, a level of adjustable coupling that will become a requirement for future highfidelity devices. This is the first solidstate demonstration of QPT in a twoqubit system, as QPT has previously been demonstrated only with single solidstate qubits^{9,10,11}.
Main
Universal quantum gates are the key elements in a quantum computer, as they provide the fundamental building blocks for encoding complex algorithms and operations. Singlequbit rotations together with the twoqubit controlledNOT (CNOT) are known to provide a universal set of gates^{12}. Here, we present the complete characterization of a universal entangling gate, the square root of iSWAP (SQiSW; ref. 13), from which gates such as the CNOT can be constructed. The pulse sequence for a CNOT can be written in terms of a SQiSW gate as follows: SQiSW R_{x}^{A}(180^{∘}) SQiSW R_{y}^{A}(90^{∘}), where R_{j}^{α}(θ) is a rotation about axis j=x,y,z by an angle θ on qubit α=A,B. The SQiSW is a ‘natural’ twoqubit gate, as it directly results from capacitive coupling of superconducting qubits, yielding qubit coupling of the general form σ_{Ax}σ_{Bx} or σ_{Ay}σ_{By}, where σ_{x,y} are the Pauli spin operators for qubits A and B (refs 14, 15). Under the rotatingwave approximation, the corresponding interaction Hamiltonian has the form H_{int}=ℏ(g/2)(01〉〈10+10〉〈01), where and g is the coupling strength that depends on design parameters.
When the two qubits are placed onresonance, the twoqubit states are coupled by H_{int} as shown in Fig. 1a. The amplitudes of these states then oscillate in time, as described (in the rotating frame) by the unitary transformation
where t is the interaction time and the representation is in the twoqubit basis set {00〉,01〉,10〉,11〉}. For an interaction time g t=π, the state amplitudes are swapped, such that 01〉→−i10〉 and 10〉→−i01〉. The SQiSW gate is formed by coupling for onehalf this time, g t=π/2, producing cosine and sine matrix elements with equal magnitudes, thus entangling the qubits. When the qubits are offresonance by an energy Δ≫g (Fig. 1b), the offdiagonal elements in U_{int} are small and have average amplitude g/Δ, effectively turning off the qubit–qubit interaction.
The electrical circuit for the capacitively coupled Josephson phase qubits^{16,17} used in this experiment is shown in Fig. 1c. Each phase qubit is a nonlinear resonator built from a Josephson inductance and an external shunting capacitance. When biased close to the critical current, the junction and its parallel loop inductance L give rise to a cubic potential with energy eigenstates that are unequally spaced. The two lowest levels are used as the qubit states 0〉 and 1〉, with transition frequency ω_{10}. This frequency can be adjusted independently for each qubit through the bias current I_{bias}^{A,B}. Each qubit’s state is detected through a singleshot measurement^{18,19}, using a fast pulse I_{Z}^{A,B} combined with readout using an onchip superconducting quantum interference device.
State preparation and tomography use singlequbit logic operations, corresponding to rotations about the x,y and z axes of the Bloch sphere^{19}. Rotations about the z axis are produced by fast (∼nanosecond) current pulses I_{Z}^{A,B}(t), which adiabatically change the qubit frequency, turning on and off the interaction and leading to phase accumulation between the 0〉 and 1〉 states. Rotations about any axis in the x–y plane are produced by microwave pulses resonant with each qubit’s transition frequency, applied through I_{μw}^{A,B}(t). The phase of the microwave pulses defines the rotation axis in the x–y plane, and the pulse duration and amplitude control the rotation angle. In previous work^{20}, such singlequbit gates were shown to have fidelities of 98%, limited by the energy relaxation T_{1} and dephasing T_{2} times, which for this device were measured to be 400 and 120 ns, respectively.
The experimental design was chosen to give qubit frequencies ω_{10}^{A,B}/2π≅5.5 GHz. The strength of the coupling g=(C_{c}/C)ω_{10}^{A,B} was set by the coupling and qubit capacitances C_{c}≈2 fF and C≈1 pF, respectively. The coupling interaction is turned on and off by changing the relative qubit frequency Δ=ω_{10}^{A}−ω_{10}^{B} through an adjustment of the qubit B bias I_{bias}^{B}. A large detuning of Δ_{off}/2π≈200 MHz was used to turn off the gate, yielding a small amplitude in the offdiagonal coupling g/Δ_{off}≈0.055.
We first characterize the coupling by measuring the time dynamics of the entangling swap operation as shown in Fig. 2a. Initially, both qubits are tuned offresonance by 200 MHz and allowed to relax to the 00〉 state. A π pulse on qubit A then produces the 10〉 state. A current pulse I_{Z}^{B}(t) applied to I_{bias}^{B} brings the qubits within a frequency Δ of resonance. After an interaction time t_{f}, the bias I_{Z}^{B} is reset to the original 200 MHz detuning and both qubits are then measured. Averaging over 1,200 events gives the probabilities for the four possible final states 00〉,01〉,10〉 and 11〉. The swapping behaviour for the states 01〉 and 10〉 as a function of t_{f} is shown in Fig. 2b. On resonance (Δ=0), the swapping frequency between 01〉 and 10〉 gives an accurate measurement of the coupling strength g/2π=11 MHz.
The amplitude of the swapping oscillations decreases with detuning as expected. In Fig. 2c we plot the peaktopeak change in swap probability as a function of detuning Δ, compared to the theoretical prediction. Apart from a small reduction in the amplitude arising from imperfect measurement fidelity, the data is in good agreement with theory. At detunings Δ/2π>50 MHz, the swap amplitude is small and cannot be distinguished from the noise floor. From the maximum detuning bias of Δ/2π=200 MHz and from the coupling strength g/2π=11 MHz obtained from spectroscopy measurements and the data in Fig. 2b, we compute the probability ratio (Δ/g)^{2}=(200/11)^{2}=300 as a figure of merit for the on/off coupling ratio.
We fully characterize the SQiSW gate using QPT^{3,4}. This involves preparing the qubits in a spanning set of input basis states, operating with the gate on this set of states, and then carrying out complete state tomography on the output. As illustrated in Fig. 3a, we first carry out quantum state tomography^{19,21} on the input state 01〉, which involves measuring the state along the x,y and z Blochsphere axes of each qubit, in nine separate experiments. We then operate on the 01〉 input state with SQiSW, and carry out complete state tomography on the output. These measurements allow for the evaluation of the twoqubit density matrix. This entire process is repeated 16 times in total, using four distinct input states for each qubit, chosen from the set {(0〉,1〉,0〉+1〉,0〉+i1〉)}. In Fig. 3b, we show the density matrix resulting from this tomography for one such input state: . From this complete set of measurements, we reconstruct the 16 by 16 χ matrix, the indices of which correspond to the Kronecker product of the operators {I,σ_{x},−iσ_{y},σ_{z}} for each qubit^{3}.
In a QPT experiment^{22,23,24} errors arise from the entangling gate and errors in measurement. As we are interested in the quality of the entangling gate itself, we have calibrated out errors resulting from measurement^{18}. As described in the Supplementary Information, measurement errors arise from both a misidentification of the 0〉 and 1〉 states, and measurement crosstalk, where a measurement of 1〉 in one qubit increases the probability of a 1〉 measurement in the second qubit^{18}. By carrying out further calibration experiments, we are able to determine the probabilities for these errors and correct the probabilities of the 00〉,01〉,10〉 and 11〉 final states.
In addition, standard QPT typically produces an unphysical χ matrix because of inherent experimental noise^{22,24}. A physical χ matrix must be completely positive and trace preserving^{3} (CPTP), which implies it must have positive eigenvalues that sum to one. Our measured χ matrix has several negative eigenvalues, as discussed in the Supplementary Information. As is commonly done in QPT experiments^{22,24}, a χ matrix that satisfies the CPTP constraints must be obtained from the experimental data before the data can be compared to a theoretically predicted χ matrix, which is physical by construction. This can be computed by noting that QPT is essentially analogous to system identification from classical control theory. More specifically, the problem of finding a physical approximation to unphysical QPT data can be shown to be a convex optimization problem^{25,26}, a technique commonly used in control theory. We use a type of convex optimization called semidefinite programming^{25} to find the physical χ matrix that best approximates our measured, unphysical χ matrix. Mathematically, for the experimentally obtained χ matrix and the physical approximation χ^{p}, we minimize the twonorm distance with the constraints that χ^{p} be CPTP.
This physical matrix χ_{m}^{p}, which also includes the calibrations for measurement errors, is shown in Fig. 4. This matrix closely matches the original data before corrections for CPTP (see Supplementary Information). In both the real and imaginary parts of the χ_{m}^{p} matrix, we observe nonzero matrix elements in locations where such elements are expected, in qualitative agreement with the theory, shown as the transparent bars. Quantitative comparison is obtained by calculating the process fidelity, 0<F_{p}<1, which gives a measure of how close χ_{m}^{p} is to theoretical expectations^{24}. For the SQiSW gate demonstrated here, with measurement calibration taken into account, we find F_{p}=Tr(χ_{t}χ_{m}^{p})=0.63, where χ_{t} is the theoretical χ matrix for the SQiSW gate. χ_{m} and χ_{e}, the unphysical χ matrices with and without measurement calibrations, respectively, and χ_{e}^{p}, the physical approximation to the χ matrix that does not include the measurement calibrations, are shown in Supplementary Figs S1–S3.
Errors in our SQiSW gate primarily arise because the time for the experiment (∼50 ns) is not significantly shorter than the T_{2} dephasing time of 120 ns. This is confirmed using a recent theory^{5} by Kofman et al., which includes the effects of dephasing and decoherence on the SQiSW χ matrix. In particular, the elements marked with an asterisk and a circle in Fig. 4 are nonzero because of energy relaxation and dephasing, respectively. The modified Pauli basis used here gives a sign change in the real part of (χ_{m}^{p})_{X X,Y Y} and (χ_{m}^{p})_{Y Y,X X}, and the imaginary part of (χ_{m}^{p})_{X X,I I} and (χ_{m}^{p})_{I I,X X}. Using this theory and the real part of (χ_{m}^{p})_{I Z,I Z} and (χ_{m}^{p})_{I Z,Z I}, we estimate our singlequbit dephasing time as T_{2}=(3π+2)/16g(χ_{m}^{p})_{I Z,I Z}. From Fig. 4 we find (χ_{m}^{p})_{I Z,I Z}=0.105 and T_{2}=123 ns, in close agreement with the value mentioned above obtained from Ramsey experiments. We also estimate the degree of correlation of the dephasing noise between the coupled qubits using κ≈(χ_{m}^{p})_{I Z,Z I}/(χ_{m}^{p})_{I Z,I Z}−[(π−2)/(3π+2)]. Our measurement of (χ_{m}^{p})_{I Z,Z I}≈0.017 yields , indicating that the dephasing is mostly uncorrelated. This is in agreement with previous work^{27,28,29} that found a dephasing mechanism local to the individual qubits.
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Acknowledgements
Devices were made at the UCSB Nanofabrication Facility, a part of the NSFfunded National Nanotechnology Infrastructure Network. We thank A. N. Korotkov for discussions on parameter extraction from the χ matrix and J. Eisert for assistance on using convex optimization to approximate a physical χ matrix. Semidefinite programming convex optimization was carried out using the opensource MATLAB packages YALMIP and SeDuMi. This work was supported by IARPA (grant W911NF0410204) and by the NSF (grant CCF0507227).
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R.C.B. designed and fabricated the samples, designed and carried out experiments and analysed the data. R.C.B. cowrote the paper with J.M.M. and A.N.C., who also supervised the project. M.A. and M.N. provided assistance with datataking software. E.L. and M.H. provided assistance with datataking electronics. M.S. provided assistance with data analysis. All authors contributed to experiment setup, sample design or sample fabrication.
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Bialczak, R., Ansmann, M., Hofheinz, M. et al. Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits. Nature Phys 6, 409–413 (2010). https://doi.org/10.1038/nphys1639
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DOI: https://doi.org/10.1038/nphys1639
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