Abstract
Quantum data are susceptible to decoherence induced by the environment and to errors in the hardware processing it. A future faulttolerant quantum computer will use quantum error correction to actively protect against both. In the smallest error correction codes, the information in one logical qubit is encoded in a twodimensional subspace of a larger Hilbert space of multiple physical qubits. For each code, a set of nondemolition multiqubit measurements, termed stabilizers, can discretize and signal physical qubit errors without collapsing the encoded information. Here using a fivequbit superconducting processor, we realize the two parity measurements comprising the stabilizers of the threequbit repetition code protecting one logical qubit from physical bitflip errors. While increased physical qubit coherence times and shorter quantum error correction blocks are required to actively safeguard the quantum information, this demonstration is a critical step towards larger codes based on multiple parity measurements.
Introduction
A recent roadmap^{1} for faulttolerant quantum computing marks a transition from storing quantum data in physical qubits to protection of logical qubits by quantum error correction (QEC)^{2,3,4,5,6} as the fourth of the seven development stages. Experimental demonstrations of QEC to date, using nuclear magnetic resonance^{7}, trapped ions^{8,9}, photons^{10}, superconducting qubits^{11} and nitrogen–vacancy centres in diamond^{12,13}, have circumvented stabilizers at the cost of decoding at the end of a QEC cycle. This decoding leaves the quantum information vulnerable to physical qubit errors until reencoding, violating a basic requirement for fault tolerance. Following steady improvements in qubit coherence, coherent control and measurement over 15 years, superconducting quantum circuits are well poised to face this outstanding challenge common to all quantum computing platforms. Initial experiments using superconducting processors include one round of either bitflip or phaseflip QEC with decoding^{11}, and the stabilization of one Bell state using dissipation engineering^{14}. Independent, parallel work^{15} demonstrates the detection of general errors on a single Bell state using stabilizer measurements.
By analogy to the classical repetition code that maps bit 0 (1) to 000 (111), the quantum version maps the onequbit superposition state α0〉+β1〉 to the entangled Greenberger–Horne–Zeilingertype (GHZ) state α0_{t}0_{m}0_{b}〉+β1_{t}1_{m}1_{b}〉 of three data qubits (labelled top, middle and bottom)^{16}. The stabilizers of this code consist of twoqubit parity measurements described by Hermitian operators Z_{t}Z_{m} and Z_{m}Z_{b}. While GHZtype states are eigenstates of both stabilizers with eigenvalue +1, their corruption by a bitflip error on one data qubit produces eigenstates with a unique pattern of −1 eigenvalues. Measuring stabilizers can thus discretize and signal single bitflip errors without affecting the encoded information (that is, the probability amplitudes α and β). Depending on the error signalled, the logical qubit is transformed to an orthogonal twodimensional subspace. It is both sufficient and better to keep track of this subspace transformation rather than attempt to return to the original subspace using correcting π pulses, which are not perfect and thus introduce errors.
In this work, we construct these stabilizers as parallelized indirect measurements using ancillary qubits, and evidence their nondemolition character by generating threequbit entanglement from the superposition states. We demonstrate stabilizerbased QEC on the minimal unit of encoded quantum information, a logical qubit, restricting to bitflip errors.
Results
Stabilizer measurements in a superconducting processor
This realization of bitflip QEC with stabilizer measurements employs a superconducting quantum processor with 12 quantum elements (Fig. 1a) exploiting resonant and dispersive regimes of circuit quantum electrodynamics^{17}. Three data transmon qubits (D_{t}, D_{m} and D_{b}) encode the logical qubit. Two ancillary transmons (A_{t} and A_{b}), two bus resonators (B_{t} and B_{b}) and two dedicated ancilla readout resonators are used for the stabilizer measurements. Dedicated readout resonators on data qubits are used to quantify performance (fidelity measures, entanglement witnessing and state tomography). All readout resonators couple to one feedline used for all qubit control and readout pulses. The feedline output couples to a single amplification chain allowing readout of all qubits by frequencydivision multiplexing^{18}. Ancilla readout fidelity is boosted by a Josephson parametric amplifier^{19} with a bandwidth covering both ancilla readout frequencies (9 MHz apart).
Building on recent developments^{20,21}, we construct quantum nondemolition stabilizer measurements in a twostep process combining entanglement with ancilla qubits and their projective measurement. Measuring the stabilizer Z_{t}Z_{m} involves an iSWAP gate between A_{t} and B_{t}, two CPHASE gates between B_{t} and each of D_{t} and D_{m}, and a final iSWAP transferring the B_{t} state onto A_{t}. These interactions correlate the joint states of D_{t} and D_{m} with even/odd (e/o) number of excitations with orthogonal states of A_{t}. Subsequently, A_{t} is measured by interrogating its dispersively coupled resonator. Conveniently, the interaction and measurement steps needed for both the stabilizers can be partially parallelized (Fig. 1c). (Note that a refocusing π pulse is applied to D_{m} after its interactions to minimize its inhomogeneous dephasing.)
We begin characterizing these stabilizer measurements by testing their ability to detect the parities of the computational states i_{t}j_{m}k_{b}〉, i,j,k∈{0,1}. Because all of these states are eigenstates of Z_{t}Z_{m} and Z_{m}Z_{b}, a fixed twobit measurement outcome P_{t}P_{b}∈{ee,eo,oe,oo} is expected for each one. Histograms of declared double parities clearly reveal the correlation (Fig. 2). The average assignment fidelity of 71%, defined as the probability of correct doubleparity assignment averaged over the eight states, is limited by errors in the interaction step. An upper bound of 91% is set by the combined readout error for the two ancilla measurements (Supplementary Table 1).
Two and threequbit entanglement by stabilizer measurements
The next test probes the ability of each stabilizer to discern twoqubit parity subspaces while preserving coherence within each. Specifically, we target the generation of two and threequbit entanglement (2QE and 3QE) via single and double stabilizer measurements on a maximal superposition state. The gate sequence in Fig. 1c is executed with D_{t} and D_{b} both prepared in and D_{m} in . First, we activate one stabilizer by performing the initial π/2 rotation only on the corresponding ancilla, and measure the dataqubitpair witness operators , (ref. 22) based on fidelity to even and oddparity Bell states, respectively. Each of these operators witnesses 2QE whenever the expectation value . With postselection on result o, either one of or witnesses 2QE at almost all values of (Fig. 3a,b). A dual result is obtained with postselection on e, for which witness entanglement (data not shown). Note that in both cases the parity of the generated entanglement differs from the detected one due to the refocusing π pulse on D_{m}.
We continue building multiqubit entanglement by activating both parity measurements and postselecting on the twobit result (Fig. 3c,d, and Supplementary Fig. 2). Ideally, P_{t}P_{b}=oo collapses the maximal superposition onto the GHZtype state . Genuine 3QE is witnessed whenever , where is the Mermin operator X_{t}X_{m}X_{b}−Y_{t}Y_{m}X_{b}−Y_{t}X_{m}Y_{b}−X_{t}Y_{m}Y_{b} (ref. 23). With postselection on P_{t}P_{b}=oo, versus reaches 2.5 (best fit, Fig. 3c). Full state tomography at the optimal reveals a fidelity 〈GHZ(0)ρGHZ(0)〉=73% to the ideal GHZ state (Fig. 3d).
This 3QEbymeasurement protocol can also be used to perform the encoding step of bitflip QEC. Ideally, the state +_{t}〉 (α0_{m}〉+β1_{m}〉) +_{b}〉 is mapped onto α1_{t}1_{m}1_{b}〉+β0_{t}0_{m}0_{b}〉 up to the transformation X_{t}X_{b}, X_{t}, X_{b}, I signalled by P_{t}P_{b}=ee, eo, oe, oo, respectively (the amplitudes α and β, in addition to the parities, are also exchanged by the refocusing π pulse on D_{m}). Postselection on P_{t}P_{b}=oo (Supplementary Fig. 3) encodes with 73% fidelity, averaged over the six cardinal input states of D_{m}, ). For comparison, implementing the standard unitary encoding^{11,24,25} using our gate toolbox (Supplementary Fig. 4) achieves 82% average fidelity.
QEC of bitflip errors
Finally, we use this encoding by gates to demonstrate bitflip QEC by parallelized stabilizer measurements (Fig. 4a). Bitflip errors are coherently added via X rotations by an angle θ, yielding a singlequbit bitflip probability p_{err}=sin^{2}(θ/2) (adding incoherent errors at this stage yields very similar results, see Methods and Supplementary Fig. 5). While the threebit code is by design only resilient to errors on a single qubit, we also consider the realistic case where such error can occur with the same probability on any of the three qubits. Therefore, we consider two scenarios: the errors added on only one data qubit and the errors added independently on all the three. In both scenarios, we assume no prior knowledge of error probability and literally interpret the stabilizer measurement results as though they were perfect. We first quantify QEC performance using the average fidelity F_{3Q} to the ideal threequbit state accounting for the subspace transformation signalled by P_{t}P_{b}=ee,eo,oe,oo (in order):
Here, is the ideal encoded cardinal state, p_{pq} is the measured probability of P_{t}P_{b}=pq, and ρ(j,pq) is the experimental pqconditioned density matrix. The near constancy of F_{3Q}(p_{err}) with errors on one qubit and the secondorder dependence with errors on all three qubits (Fig. 4b) reflect the ability of the stabilizers to discretize and signal singlequbit bitflip errors without decoding.
To assess the ability of QEC to detect added errors without unfairly penalizing for intrinsic decoherence and encoding errors, we compare F_{3Q} with the stabilizer interactions replaced by idling for equal duration (with a refocusing D_{m} pulse):
Without QEC, one expects a linear decrease in F_{3Q} with errors on one qubit as one bit flip orthogonally transforms the encoded state. The slight curvature observed reflects residual coherent errors in encoding. The nonmonotonicity of F_{3Q} with errors on all qubits reflects that triple errors perform a logical bit flip, which leaves +_{L}〉 and −_{L}〉 unchanged. Comparing the curves suggests that QEC provides net gains for p_{err}≳15% in the first case and for p_{err}≳10% in the second (Fig. 4b).
Discussion
However, the true merit of QEC hinges on the ability to suppress the accumulation of errors. We believe that a better comparison is the logical state fidelity F_{L} following two rounds of errors with QEC or idling in between. F_{L} is defined as the fidelity to the initial unencoded D_{m} state following an ideal decoder (Fig. 4a) that is resilient to a bitflip error remaining in any one qubit. For example, with QEC and a secondround error ,
Here we consider the scenario with errors on all three qubits and only incoherent secondround errors. We expect QEC to win over idling in select cases, such as single errors on both rounds but on different qubits, all of which we observe (Fig. 4d and also Supplementary Fig. 6). Weighing in all the possible cases (from 0 to 3 errors in each round) according to their probability, we find that the current fidelity of the stabilizer measurements precludes boosting F_{L} for the cardinal states using this quantum repetition code at any p_{err} (Fig. 4c). This stricter comparison sets the benchmark for gauging future improvements in QEC.
In summary, we have realized parallel stabilizer measurements with ancillary qubits and used them to perform a quantum repetition code on a superconducting circuit. Stabilizerbased QEC can detect bitflip errors on data qubits while maintaining the encoding at the logical level, thus meeting a necessary condition for faulttolerant quantum computing. Evidently, it remains a priority to extend qubit coherence times and shorten the QEC step to boost logical fidelity. In the longer term, parallelized ancillabased parity measurements as demonstrated here may be used to protect a logical qubit against general errors with a Steane^{6,26} or small surface code^{27}.
Methods
Processor fabrication
The integrated circuit is fabricated on a cplane sapphire substrate. A NbTiN film (80 nm) is reactively sputtered at 3 mTorr in a 5% N_{2} in Ar atmosphere, resulting in a superconducting critical temperature of 15.5 K and normalstate resistivity of 110 μΩcm. This film is ebeam patterned using SAL601 resist and etched by SF_{6}/O_{2} RIE to define all coplanar waveguide structures: feedline, resonators and fluxbias lines. The transmon Josephson junctions and shunting interdigitated capacitors are patterned using PMGI/PMMA ebeam lithographed resist and doubleangle shadow evaporation of Al with intermediate oxidization. Air bridges are added to suppress slotline propagation modes, to connect ground planes and to allow the crossing of transmission lines (Supplementary Fig. 8). Bridge fabrication starts with a 6μmthick PMGI layer, which is patterned and then reflowed at 220 °C for 5 min, producing a gently arched profile. A second MAA/PMMA resist layer is spun and ebeam patterned to define the bridge geometry. Finally, Ti (5 nm) and Al (450 nm) are ebeam evaporated. The 2 mm by 7 mm chip is diced and cleaned in 88 °C NMP for 30 min.
Experimental setup
The quantum processor is anchored to the mixing chamber plate of a dilution refrigerator with 15–20 mK base temperature. A detailed schematic of the experimental setup at all temperature stages is shown in Supplementary Fig. 8. The single coaxial line for readout and microwave control has inline attenuators and absorptive lowpass filters providing thermalization, noise reduction and infrared radiation shielding. Coaxial lines for flux control are broadband attenuated and bandwidth limited (1 GHz) with reactive and absorptive lowpass filters.
Qubit control
Most microwave pulses for X and Y qubit rotations have a Gaussian envelope in the main quadrature (5 ns sigma and 20 ns total duration), and a derivativeofGaussian envelope in the other (DRAG pulses^{28}). Wah–Wah pulses^{29} combining DRAG with sideband modulation are used for D_{t} and A_{b} to avoid leakage in D_{m} and D_{b}, respectively. Taking advantage of the proximity in frequency between D_{t} and A_{t}, and between D_{m} and A_{b}, we coherently control the five qubits by sideband modulation of three carriers (Supplementary Fig. 8).
Flux pulses for iSWAPs are sudden (12 ns duration), while those for CPHASEs are mostly fast adiabatic^{30} (40 ns). The pulse for CPHASE between D_{m} and B_{t} is kept sudden (19 ns) to avoid leakage during the crossing of D_{m} through B_{b}. Pulse distortion resulting from the flux control bandwidth is minimized by manual optimization of convolution kernels.
Qubit readout
The five qubits are readout by frequency division multiplexing^{18}. The readout pulses for data and ancilla qubits are separately generated by sideband modulation of the two carriers.
The amplitude and duration of readout pulses are chosen to maximize assignment fidelity. D_{t}, D_{m} and D_{b} readout pulses have 1,200, 1,000 and 700 ns duration, respectively. The signaltonoise boost provided by the Josephson parametric amplifier allows shorter ancilla qubit readouts, 600 ns (550 ns) for A_{t} (A_{b}). The amplified feedline output is split and downconverted with two local oscillators. The two signals are amplified, digitized, demodulated and integrated to yield one voltage for each qubit measured. The low crosstalk between the qubit readouts is evidenced by simultaneous measurement immediately following preparation of the 32 combinations of the five qubits in either 0〉 or 1〉 (Supplementary Fig. 9).
Using the method of ref. 20 based on Hahn echo sequences, we have bound the dephasing of each data qubit induced by the ancilla measurements to <1% (data not shown). Since dataqubit fidelity loss during ancilla measurements is dominated by intrinsic decoherence and our main interest is to quantify the ability of stabilizers to detect the intentionally added errors, we have opted to advance the data qubit measurements, making them simultaneous to those of ancillas (Supplementary Fig. 4).
Initialization
The four qubits {D_{t}, D_{b}, A_{t} and A_{b}} and two buses {B_{t} and B_{b}} are initialized to their ground state by postselection on six measurements performed before any encoding or manipulation protocol. The buses are initialized by swapping states with their coupled ancilla immediately after initialization of the latter. D_{m} is initialized by swapping its excitation (∼10%) with that of B_{b} (∼1%). The postselected fraction of runs (50–60%) have a residual excitation of 1–2% in every quantum element.
Gate sequence
Gates are parallelized as much as possible. We note two important exceptions. Because of frequency crowding and the common feedline, pulses targeting one qubit induce ac Stark shifts on untargeted qubits. We serialize singlequbit control to restrict the effect of these shifts to residual phase rotations on unaddressed qubits. Also, the first iSWAP between B_{t} and A_{t} and CPHASE between B_{t} and D_{m} (Fig. 1c) are applied before populating B_{b} to avoid a strong dispersive shift of D_{m}. All others iSWAPS, CPHASE gates and ancilla measurements are simultaneous.
Incoherent errors
We have also tested stabilizerbased QEC with incoherent firstround errors generated using π rotations (Supplementary Fig. 5). Following encoding of a D_{m} cardinal input state , we apply the eight combinations of error/no error on the three data qubits. We calculate F_{3Q} and F_{L} for each combination and weigh by the corresponding probability.
Additional information
How to cite this article: Ristè, D. et al. Detecting bitflip errors in a logical qubit using stabilizer measurements. Nat. Commun. 6:6983 doi: 10.1038/ncomms7983 (2015).
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Acknowledgements
We thank K.W. Lehnert for providing the parametric amplifier, D.J. Thoen and T.M. Klapwijk for sputtering of NbTiN films, K.M. Svore, T.H. Taminiau, D.P. DiVincenzo, E. Magesan and J.M. Gambetta for fruitful discussions, and N.K. Langford, G. de Lange, L.M.K. Vandersypen and R. Hanson for helpful comments on the manuscript. We acknowledge the funding from the Netherlands Organization for Scientific Research (NWO), the Dutch Organization for Fundamental Research on Matter (FOM) and the EU FP7 project ScaleQIT.
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A.B. fabricated the processor, with design input from O.P.S. and L.D.C. O.P.S. and V.V. performed the initial tuneup. D.R., M.Z.H. and S.P. performed measurements and data analysis. S.P., D.R. and L.D.C. prepared the manuscript with feedback from all the other authors. L.D.C. supervised the project.
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Supplementary Figures 19 and Supplementary Table 1 (PDF 22064 kb)
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Ristè, D., Poletto, S., Huang, MZ. et al. Detecting bitflip errors in a logical qubit using stabilizer measurements. Nat Commun 6, 6983 (2015). https://doi.org/10.1038/ncomms7983
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DOI: https://doi.org/10.1038/ncomms7983
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