Abstract
The ability to detect and deal with errors when manipulating quantum systems is a fundamental requirement for faulttolerant quantum computing. Unlike classical bits that are subject to only digital bitflip errors, quantum bits are susceptible to a much larger spectrum of errors, for which any complete quantum errorcorrecting code must account. Whilst classical bitflip detection can be realized via a linear array of qubits, a general faulttolerant quantum errorcorrecting code requires extending into a higherdimensional lattice. Here we present a quantum error detection protocol on a twobytwo planar lattice of superconducting qubits. The protocol detects an arbitrary quantum error on an encoded twoqubit entangled state via quantum nondemolition parity measurements on another pair of error syndrome qubits. This result represents a building block towards larger lattices amenable to faulttolerant quantum error correction architectures such as the surface code.
Introduction
Errors are inevitable in any real information processor. Quantum computers are particularly susceptible to errors as quantum systems are highly sensitive to noise effects that can be exotic compared with the simple bitflip errors of classical computation. As such, realizing a faulttolerant quantum computer is a significant challenge that requires encoding the information into a quantum errorcorrecting code. To add to the difficulty, direct extraction of the information typically destroys the system, and ancillary syndrome systems must be employed to perform nondemolition measurements of the encoded state. Previous work in nuclei^{1,2,3}, trapped ions^{4,5,6} and superconducting qubits^{7} has attempted to address similar problems; however, these implementations lack the ability to perform faulttolerant syndrome extraction, which continues to be a challenge for all physical quantum computing systems.
The surface code (SC)^{8,9} is a promising candidate to achieve scalable quantum computing due to its nearestneighbour qubit layout and high faulttolerant error thresholds^{10}. The SC is an example of a stabilizer code^{11}, which is a code whose state is uniquely defined by the measurement of a set of observables called stabilizers. Code qubits in the SC are placed at the vertices of a twodimensional array and each stabilizer involves four neighbouring code qubits. The SC stabilizers are, therefore, geometrically local and can be measured fault tolerantly with a single syndrome qubit^{12}. Error detection on a lattice of code qubits is achieved through mapping stabilizer operators onto a complementary lattice of syndrome qubits, followed by classical correlation of measured outcomes. Among the syndrome qubits, a distinction is made between bitflip syndromes (or Zsyndromes) and phaseflip syndromes (or Xsyndromes). Each code qubit in the SC is coupled with two Xsyndrome qubits and two Zsyndrome qubits, and, in turn, each syndrome qubit is coupled with four code qubits.
Superconducting qubits have become prime candidates for SC implementation^{13,14}, especially with continuing improvements to coherence times^{15,16,17} and quantum gates^{18}. Furthermore, implementing superconducting resonators as quantum buses to realize the circuit quantum electrodynamics architecture permits a straightforward path for building connectivity into a lattice of superconducting qubits^{14}. There are numerous ways of building the SC lattice with superconducting qubits and resonators. Here we employ an arrangement in which each qubit is coupled with two bus resonators and each bus couples with four qubits^{14}. Although previously the engineered dissipation of a resonator has been used to stabilize the entanglement of two superconducting qubits to which it is coupled^{19}, it is of note that here the stabilization is achieved via explicitly mapping code qubit stabilizers onto syndrome qubits.
Here we experimentally demonstrate the complete algorithm constituting a quantum error detection code that detects arbitrary singlequbit errors in a nondemolition manner via syndrome measurements. The scheme is implemented in a twobytwo lattice of superconducting qubits that represents a primitive tile for the SC. Stabilizer measurements, ubiquitous to faulttolerant quantum errorcorrecting codes, are successfully demonstrated in this work for both bit and phaseflip errors on an encoded codeword. The nondemolition nature of the protocol is verified by demonstrating the preservation of the entangled state constituting the codeword through highfidelity syndrome measurements in the presence of an arbitrary applied error. These error detection experiments constitute a key milestone for SC implementation, as our operations now extend into the plane of the twodimensional surface and we show the ability to concurrently perform bit and phaseparity checks. Moreover, our results illustrate the ability to build structures of superconducting qubits, which are not colinear but latticed while preserving highfidelity operations. Moving forward, on improving the measurement and gate fidelities in these systems, further expanding the lattice will lead to important studies of different errorcorrecting codes and the encoding of logical qubits, thereby allowing experimental investigation of faulttolerant quantum computing. Our results bolster the prospect of employing superconducting qubit lattices for largescale faulttolerant quantum computing.
Results
Physical device and quantum control
Our physical device (Fig. 1a,b) consists of a 2 × 2 lattice of superconducting transmons, with each coupled with its two nearest neighbours via two independent superconducting coplanar waveguide (CPW) resonators serving as quantum buses (Fig. 1b; blue). Each qubit is further coupled with an independent CPW resonator for both qubit control and readout. Dispersive readout signals for each qubit are amplified by distinct Josephson parametric amplifiers (JPAs) giving high singleshot readout fidelity^{20,21}. We implement twoqubit echo crossresonance (ECR) gates^{22}, ECR=ZX_{90}–XI, which are primitives for constructing controlledNOT (CNOT) operations. Given the latticed structure of our device, we implement four different such gates, ECR^{ij} between qubits Q_{i} (control) and Q_{j} (target), with ij∈{12,23,34,41}. In this arrangement, we use Q_{1} and Q_{3} (Fig. 1b; purple) as code qubits, Q_{2} as the Zsyndrome qubit (Fig. 1b; green) and Q_{4} as the Xsyndrome qubit (Fig. 1b; yellow). All ECR gates are benchmarked^{22} with fidelities between 0.93 and 0.97 (for further details see Methods). Singlequbit gates are benchmarked to fidelities above 0.998 with <0.001 reduction in fidelity due to crosstalk, as verified via simultaneous randomized benchmarking^{23}. The four independent singleshot readouts yield assignment fidelities (see Methods) all above 0.94. These and other relevant system experimental parameters including qubit frequencies, anharmonicities, energy relaxation and coherence times are further discussed in the Methods.
Quantum error detection protocol implementation
Our fourqubit square lattice is a nontrivial cutout of the SC layout (circled in Fig. 1a), and can be used to demonstrate both the ZZ and XX parity check. The XX (ZZ) stabilizer is measured by the Xsyndrome (Zsyndrome) qubit. Although previous work^{14,24,25,26} implemented parity checks on linear arrangements of qubits, our experiment goes beyond into the other planar dimension. The extra dimension allows us to demonstrate the [[2,0,2]] code, which contains 2 physical qubits, 0 logical qubits (and hence a single fixed code state), and has a distance of 2, which means arbitrary singlequbit errors are detectable. The codeword is the twoqubit entangled state , which is protected from any singlequbit error on the codespace via syndrome detection. An arbitrary singlequbit error revealed in the stabilizer syndrome as a bit (phase) flip simply maps to a negative eigenstate of ZZ (XX), and a joint bit and phaseflip (Y rotation) maps to the negative eigenstate of both ZZ and XX. By encoding both the XX and the ZZ stabilizers in the fourqubit lattice, we can protect a maximally entangled state of the twocode qubits against an arbitrary error.
To demonstrate the SC sublattice stabilizer measurement protocol (Fig. 1c), we first prepare the twocode qubits in codeword state , which is a maximally entangled Bell state. Subsequently, the ZZ stabilizer is encoded onto the Zsyndrome qubit Q_{2}, which is initialized in the ground state . The XX stabilizer is encoded onto the Xsyndrome qubit Q_{4}, which is initialized in the state. Since we perform measurements of the syndrome qubits in the Z measurement basis, Q_{4} also undergoes a Hadamard transformation H right before measurement. The complete circuit as shown in Fig. 1c will detect an arbitrary singlequbit error ɛ to the code qubits via the projective measurements of the syndrome qubits. We choose to apply the error on Q_{1}, but there is no loss of generality if applied on Q_{3} instead. Each of the four possible outcomes of the syndrome qubit measurements projects the code qubits onto one of the four maximally entangled Bell states. If no error is present in the sequence, the syndrome qubits are both found to be in their ground state after the measurement, and the prepared codeword state of the code qubits is preserved.
In our experiment, since the twocode qubits (Q_{1} and Q_{3}) are nonnearest neighbours in the lattice, the preparation of the codeword state is performed via twoqubit interactions with a shared neighbouring qubit, Q_{2}. The gate sequence for this state preparation can be compiled together with portions of the ZZ stabilizer encoding. The resulting complete gate decomposition of the circuit from Fig. 1c in terms of our available single and twoqubit ECR gates is described in detail in the Methods.
To implement arbitrary errors to the entangled code qubit state, we apply singlequbit rotations to Q_{1} of the form ɛ=U_{θ}, where U defines the rotation axis and θ is the rotation angle (when no angle is given it is assumed θ=π). Following the error detection protocol of Fig. 1c, we acquire singleshot measurements of the syndrome qubits and correlate independent measurements around various axes of the code qubits for quantum state tomography^{27}. First, for the case where ɛ=U_{0}, when no error is added, the two syndrome qubits Q_{2} and Q_{4} should both be measured to be in their ground states, and from correlating their singleshot measurements, M_{2} and M_{4}, we detect the colour map as shown in Fig. 2a. Here, we can clearly see that a majority of the resulting measurements are located in the lower left quadrant, and we will use the notation {M_{2},M_{4}}={0,+}, with both syndromes signalling a ground state detection (note that measuring Q_{4} in the ground state signals a detection given the H before measurement). Conditioned on {0,+}, state tomography of the code qubits is performed, with a reconstructed final state (Pauli vector shown in Fig. 2a), commensurate with the originally prepared codeword state with a fidelity of 0.8491±0.0005. Next, for the case of a bitflip error to Q_{1}, or ɛ=X_{π}, the resulting syndrome histograms are shown in the colour map in Fig. 2b, where a majority of results are consistent with {1,+}, where the Zsyndrome Q_{2} is excited to and the Xsyndrome Q_{4} remains in its ground state. Conditioned on {1,+}, the reconstructed final state Pauli vector of the code qubits is now , verifying the bitflip parity error. Then, for the case of a phaseflip error on Q_{1}, or ɛ=Z_{π}, we find that the syndromes give {0,−}, with the Xsyndrome having changed its state (Fig. 2c). Similarly, conditioned on {0,−}, the code qubit state agrees with , showing the phase flip. Finally, an error ɛ=Y_{π} results in both syndromes flipped, {1,−}, as shown in Fig. 2d with corresponding code qubit Pauli vector in agreement with both a bit and phase flip of the original codeword state .
The reconstructed states reveal important information about our system. First, the measured state fidelity (∼0.80–0.84) is higher than expected (∼0.75) from the measured fidelities of the five twoqubit gates and two independent singleshot measurements. This is because the gates used to prepare the codeword state do not contribute to the accumulated state fidelity loss, but rather reveal themselves as measurement errors. Second, the reconstructed conditional states have little to no weight in the singlequbit subspace. This suggests that in our system there are negligible crosstalk errors (as expected since the code qubits are not directly connected via a bus).
Tracking arbitrary errors
We can track the outcome of the syndrome qubits as we slowly vary θ in an applied error ɛ=Y_{θ} between −π and +π (see Fig. 3). The state population of the four syndrome qubit states, {0,+} (black dots), {1,+} (red dots), {0,−} (green dots) and {1,−} (blue dots), obtained from the (normalized) number of counts in the correlated histograms conditioned on a readout threshold extracted from calibration measurements (see Methods), are plotted versus θ. For an error induced by a unitary operation, the data is explained by cosines (solid lines in Fig. 3). For θ near 0, the ground state, {0,+}, is found in both syndrome qubits as expected, whereas for θ∼π, we recover both the syndrome flips {1,−}. The observed contrast between the different syndrome qubit state populations, near 0.6 in Fig. 3, is commensurate with a masterequation simulation that takes into account the measured coherence times of our qubits and the assignment fidelities of the readouts (dashed lines in Fig. 3). Similarly, varying θ for X and Z rotations are shown in the Methods.
To demonstrate arbitrary error detection, we construct ɛ via combinations of X and Y errors. Each panel of Fig. 4 shows a teal bar plot reflecting the experimentally extracted population of each of the four possible syndrome qubit measurement outcomes for the set of errors {Y_{π/3}, X_{π/3}, X_{π/3}Y_{π/3}, X_{π/3}Y_{2π/3}, X_{2π/3}Y_{π/3}, X_{2π/3}Y_{2π/3}, R, H}, where R=Y_{π/2}X_{π/2} and H is the Hadamard operation. Overall, we find decent agreement between the experiment and ideal population outcomes (dark blue bars). The measured populations are renormalized by the observed contrast at θ∼0 in Fig. 3, and in the equivalent plots for X and Z errors shown in the Methods, to account for relaxation and decoherence fidelity loss. Although this renormalization provides an overall fairer comparison with the ideal case, it tends to increase the uncertainty in the bars, especially for the Z error due to the loss of contrast and larger data scatter observed for the Xsyndrome qubit Q_{4} (see Methods). This diminished contrast is due to the fact that the XX stabiliser is encoded last in our sequence and therefore suffers more from decoherence.
Discussion
We have provided a set of experiments that demonstrate the detection of arbitrary singlequbit quantum errors on a square lattice of qubits. The experiments combine a variety of key components required for scaling quantum systems up to larger numbers of qubits: highfidelity one and twoqubit gates, high singleshot assignment fidelities allowing for nondemolition measurements of code qubits and improved system design to minimize crosstalk effects in nontrivial lattices of nearestneighbourcoupled qubits. Moving forward, continued improvement of gate and assignment fidelities will be required to reach faulttolerance thresholds. This will require continued understanding of potential sources of error in our system, such as calibration and crosstalk effects, as well as improved system design and engineering. In addition, achieving shorter measurement times and measurement repeatability will be key for demonstrating largescale experimental quantum error correction. While there remain significant challenges in implementing complex operations on larger lattices of qubits, the work we have presented here demonstrates that a high degree of control and microwave hygiene can be achieved with superconducting qubits arranged in geometries useful for faulttolerant quantum computation.
Methods
Device fabrication and parameters
The device is fabricated on a 720μmthick Si substrate. The superconducting CPW resonators, the qubit capacitors and coupling capacitors are defined in the same step via optical lithography. Reactive ion etching of a sputtered 200nmthick Nb film is used to make this layer. The Josephson junctions, patterned via electron beam lithography, are made by doubleangle deposition of Al (layer thicknesses of 35 and 85 nm) followed by a liftoff process. The chip is mounted on a printed circuit board and wirebonded for signal delivery and crosstalk mitigation.
The fourqubit transition frequencies are ω_{i}/2π={5.303,5.101,5.291,5.415} GHz with i∈{1,2,3,4}. The readout resonator frequencies are ω_{Ri}/2π{6.494,6.695,6.491,6.693} GHz, while the four bus resonators, unmeasured, are designed to be at ω_{Bii}/2π={8,7.5,8,7.5} GHz for ij∈{12,23,34,41}. All qubits show around 330 MHz anharmonicity, with energy relaxation times T_{1(i)}={33,36,31,29} μs and coherence times μs. The dispersive shifts and line widths of the readout resonators are measured to be 2χ_{i}/2π={−3.0,−2.0,−2.5,−2.8} MHz and κ_{i}/2π={615,440,287,1210} kHz, respectively.
Gate calibration and characterization
Singlequbit gates are 53.3ns long Gaussian pulses with width σ=13.3 ns. We use singlesideband modulation to avoid mixer leakage at the qubit frequencies in between operations. The sideband frequencies, which are chosen taking into account all qubit frequencies and anharmonicities, are +60, −80, +180 and +100 MHz for Q_{1}, Q_{2}, Q_{3} and Q_{4}, respectively. Every singlequbit pulse is accompanied by a scaled Gaussian derivative in the other quadrature to minimize the effect of leakage of information into higher qubit energy levels^{28}. All microwave mixers are independently calibrated at the operational frequencies to minimize carrier leakage as well as to ensure orthogonality of the quadratures. Following these calibrations, the singlequbit rotations are tuned by a series of repeated rotations described elsewhere^{29}.
Randomized benchmarking (RB) of singlequbit gates^{30} is performed for all four qubits independently and in all possible simultaneous configurations (Table 1). This allows us to establish the degree of addressability error^{23} present in our system. Comparing the individual and simultaneous RB experiments, we can see that the addressability error is 0.001 or lower in all cases.
The twoqubit ECR gates consist of two crossresonance pulses of different signs, each of duration τ, separated by a π rotation in the control qubit. This sequence selectively removes the IX part of the Hamiltonian while enhancing the ZX term^{22}. Each crossresonance pulse has a Gaussian turnon and off of width 3σ with σ=24 ns, included in τ. The gates ECR^{12}, ECR^{23}, ECR^{34} and ECR^{41}, where ECR^{ij} is the ECR gate between Q_{i} (control) and Q_{j} (target), had τ of 400, 360, 440 and 190 ns, respectively, for a total gate time of 2 × τ+53.3 ns. We also characterise the twoqubit gates via Clifford RB^{22}. Figure 5 shows the RB decays for each of the four gates, yielding an error per twoqubit Clifford gate of 0.0604±0.0006, 0.0631±0.0007, 0.0569±0.0015 and 0.0353±0.0015 for ECR^{12}, ECR^{23}, ECR^{34} and ECR^{41}, respectively.
Experimental setup
We cool our device to 15 mK in an Oxford Triton dilution refrigerator. Figure 6 shows a full schematic of the measurement setup. We achieve independent singleshot readout for each qubit using a highelectronmobility transistor (HEMT) amplifier following a JPA (provided by UC Berkeley) in each readout line. The device is protected from environmental radiation by an Amuneal cryoperm shield with an inner coat of Emerson & Cuming CR124 Eccosorb. All qubit control lines are heavily attenuated at different thermal stages and homemade Eccosorb microwave filters are added at the coldest refrigerator plate. Figure 7 shows the circuit schematic at chip level, including the design of the qubit capacitance and coupling lines.
Singlequbit and twoqubit control pulses as well as resonator readout pulses are generated using singlesideband modulation. The modulating tones are produced by Tektronix arbitrary waveform generators (model AWG5014) for qubit operations. Modulating shapes for readout are produced by Arbitrary Pulse Sequencers from Raytheon BBN Technologies. We either use external Marki I/Q mixers with a Holzworth microwave generator or an Agilent vector signal generator (E8257D) as depicted in Fig. 6. For data acquisition, we use two AlazarTech twochannel digitizers (ATS9870) and the singleshot readout time traces are processed with an optimal quadrature rotation filter^{27}.
Circuit gate decomposition
The circuit in Fig. 1c calls for four twoqubit gates. In addition, the code qubits Q_{1} and Q_{3} need to be prepared in an entangled state. Since these qubits are not nearest neighbours and there is no provision for interaction between them—a key feature of the SC—we first entangle Q_{1} and Q_{2} and then perform a swap operation between Q_{2} and Q_{3} (Fig. 8a). A SWAP gate operation is equivalent to three CNOT gates alternating direction (Fig. 8b). Since two consecutive identical CNOT gates are equal to the identity operation and Q_{3} starts from the ground state, the red shadowed regions in Fig. 8 can be omitted. The actual circuit implemented in our experiments is shown in Fig. 8c, where the Bell state preparation and the ZZ encoding have been combined.
Our CNOT operations require an entangling gate between the control and the target qubits. We use the ECR^{ij} as our CNOT genesis. The ECR^{ij} gate plus four singlequbit rotations as depicted in Fig. 8c correspond to a CNOT operation between Q_{i} and Q_{j} in our device.
Tracking bit and phaseflip errors
As introduced in the main text (Fig. 3), we can measure the magnitude of the error ɛ from the correlated singleshot traces of the syndrome qubits. Here we show the figures complementing Fig. 3 in the main text, corresponding to pure bitflip error (Fig. 9a) and pure phaseflip error (Fig. 9b). We attribute the increased loss of contrast in the phaseflip error detection (Fig. 9b) to the order of the stabilizer encoding in our circuit, which makes our error detection protocol less sensitive to phaseflip errors.
Error propagation and syndromes
After the SWAP gate and error in the circuit in Fig. 8a, the state of the code qubits is given by , where ɛ is some unitary operator acting on the firstcode qubit. We will find how the different Pauli errors propagate through the rest of the circuit to produce the different error syndromes.
First, suppose ɛ is the bitflip operation on the firstcode qubit C_{1}. In this case,
where the subindexes S_{1} and S_{2} refer to the Z and Xsyndrome qubits, Q_{2} and Q_{4} in our experiment, respectively. Similarly for the phaseflip operation ,
and for
Since the state after the SWAP gate is (the qubits are ordered and Q_{1}, Q_{3} are the code qubits), the error syndromes are given by
Hence, if the error is a general singlequbit unitary operation
the different error syndromes have the following probabilities of occurring
where n_{i} is the i^{th} component of the unit vector.
Readout characterization
To characterize each readout, we create the 2^{4}=16 standard computational basis (calibration) states and record the full timedependent trajectory of the state of the cavity over a measurement integration time of 3 μs. This process is repeated 19,200 times to gather sufficient statistics. Integrating kernels are obtained for each measurement channel, which extract the full timedependent readout information^{27}. Histograms are fitted to the integrated shots and thresholds for each channel are set at the point of maximum distance between cumulative distributions of the histograms.
The assignment fidelity of each channel is calculated according to the standard formula
where P(01) (P(10)) is the probability of obtaining ‘0’ (‘1’) when state () is created. The assignment fidelities are given in Supplementary Table 1.
State tomography
The conditional states of the code qubits (Q_{1} and Q_{3}) for the different error types (I, X, Y and Z) were reconstructed by applying the complete set of 36 unitary rotations to the state of code qubits to attain a complete set of measurement operators. The fundamental measurement observables that are rotated by elements of are constructed from the calibration states by first normalizing the shots for each of the code qubit channels to lie in [−1,1] and then correlating the shots. Note that if the calibrations were perfect, then are equal to ZI, IZ and ZZ.
For each of the 36 different measurement settings, we bin each shot according to the measurement results of the syndrome qubits Q_{2} and Q_{4}. As there are two syndrome qubits, there are four bins labelled by down–down, down–up, up–down and up–up. Denoting the conditional states ρ^{dd}, ρ^{du}, ρ^{ud} and ρ^{uu} and we have full tomographic information of the state of the code qubits for each of the four bins. The shots are correlated to create the expectation values of each conditional state. Hence, for each , label ab, and observable , we have an estimate of trace.
For each label ab, we have a measurement vector m^{ab} of length 108 (36 unitary rotations × 3 fundamental observables). Choosing any representation x^{ab} of ρ^{ab} in some operator basis allows us to write
where M is a constant matrix whose entries depend only on the choice of operator basis. We choose to use the standard Pauli basis to represent ρ^{ab},
which implies M is a 108 × 16 matrix. Enforcing ρ^{ab} to be trace 1 sets x_{0}=1/4.
x^{ab} can be solved for in a variety of ways, the most straightforward of which is linear inversion via computing the pseudoinverse of M. While linear inversion provides a valid statistical estimator, it does not enforce positivity of the state. Alternatively, one can maximize the likelihood function for the measurement results under the assumption of Gaussian noise^{29} and solve the following constrained quadratic optimization problem
to obtain a physically valid state. Here V^{ab} is the variance matrix of the measurement matrix. When only Gaussian noise is present, solving this optimization problem is equivalent to finding the closest physical state to the linear inversion estimate^{31}.
We quantify the state reconstruction via the state fidelity between ρ_{noisy}=ρ^{ab} and the ideal target state ;
where, as mentioned in the main text, the ideal states for the different syndrome results are given by
The results are contained in Supplementary Table 2. The variance in the state fidelity is computed via a bootstrapping protocol described in ref. 29 and the physicality is the sum of the negative eigenvalues of the linear inversion estimate. We see that linear inversion produces physical estimates in all cases and there is negligible difference between the fidelities of the physical and linear inversion estimates.
Insensitivity to statepreparation errors
Since we are conditioning on the measurement results of the syndrome qubits, the error detection circuit has the useful feature that rotation errors on the prepared (encoded) twoqubit state correspond only to decreasing the success probability of preserving the desired state. From a tomographic standpoint, we can accurately reconstruct the conditioned state as long as the total number of shots is large relative to the error syndrome probability, so that sufficient measurement statistics are available.
To make this precise, suppose that the ideal initial state is rotated via some error operator E to the state
This gives the following syndrome probabilities:
and so the probability of successfully obtaining the correct state is a^{2}. Since the shots producing the error syndromes are evenly distributed throughout the different unitary rotation pulses on the code qubits, the effect on the code state is to reduce the number of shots by a factor of a^{2}, which can also be thought of as a rescaling of the measurement variances by . Hence, for each of the 108 different measurement observables , has a singleshot variance that scales as . This implies we expect that, to first order in θ, state tomography is robust to overunder rotation errors.
We can model and verify this effect by directly applying a unitary error of varying strength on the firstcode qubit. The general unitary Kraus operator is and the probabilities for the different syndromes are given by equation (6). For simplicity, we chose a purely X rotation so ɛ=cos(θ)I−i sin(θ)X and varied the size of the angle in 30 steps from −π to π. The state fidelity as a function of θ is shown in Supplementary Fig. 1. As expected, the first derivative appears to smoothly converge to 0 as θ converges to 0 and the loss in fidelity is a result of insufficient statistics for the 00syndrome state.
This discussion also allows us to more accurately predict the output conditional state fidelities. As demonstrated, we can effectively ignore coherent errors in the first two CNOT gates, since they are used for state preparation and errors in these operations show up as a reduction in the number of shots available for tomography. Assuming the number of shots is large enough, and ignoring singlequbit errors, we are only concerned with errors in the final three CNOT gates. From twoqubit RB, the average gate fidelity of our CNOT gates is ∼0.94. Hence, assuming depolarizing errors, we can obtain an approximate gate fidelity for the comprised circuit of ∼0.94^{3}=0.83 and state fidelities with similar values, which is consistent with our obtained fidelities in Supplementary Table 2.
Additional information
How to cite this article: Córcoles, A.D. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6:6979 doi: 10.1038/ncomms7979 (2015).
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Acknowledgements
We thank M. B. Rothwell and G. A. Keefe for fabricating devices. We thank J. R. Rozen, J. Rohrs and K. Fung for experimental contributions. We thank S. Bravyi and J. A. Smolin for engaging discussions. We thank I. Siddiqi for providing the JPAs. We acknowledge Caltech for HEMT amplifiers. We acknowledge support from IARPA under contract W911NF1010324. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of the US Government.
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J.M.C. and J.M.G. designed the experiments. A.D.C. and S.J.S. characterized devices and ran the experiments. A.W.C, J.M.G. and M.S. developed the gate breakdown for the code implementation. E.M., A.D.C. and J.M.G interpreted and analysed the experimental data. All authors contributed to the composition of the manuscript.
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Correspondence to A.D. Córcoles.
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Supplementary Figure 1 and Supplementary Tables 12 (PDF 88 kb)
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Córcoles, A., Magesan, E., Srinivasan, S. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat Commun 6, 6979 (2015). https://doi.org/10.1038/ncomms7979
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