Demonstration of a quantum error detection code using a square lattice of four superconducting qubits

Abstract

The ability to detect and deal with errors when manipulating quantum systems is a fundamental requirement for fault-tolerant quantum computing. Unlike classical bits that are subject to only digital bit-flip errors, quantum bits are susceptible to a much larger spectrum of errors, for which any complete quantum error-correcting code must account. Whilst classical bit-flip detection can be realized via a linear array of qubits, a general fault-tolerant quantum error-correcting code requires extending into a higher-dimensional lattice. Here we present a quantum error detection protocol on a two-by-two planar lattice of superconducting qubits. The protocol detects an arbitrary quantum error on an encoded two-qubit entangled state via quantum non-demolition parity measurements on another pair of error syndrome qubits. This result represents a building block towards larger lattices amenable to fault-tolerant quantum error correction architectures such as the surface code.

Figure 1. Surface code implementation and error detection quantum circuit.

(a) Cartoon schematic of SC consisting of alternating square tiles of X- (yellow) and Z- (green) plaquettes for detecting phase-flip (Z) and bit-flip (X) errors, respectively. Semi-circular pieces reflect parity checks at the boundaries of the lattice. These plaquette tiles can be mapped onto a lattice of physical superconducting qubits with appropriate nearest-neighbour interconnectivity, as shown in the layer labelled MAP. Here there are code qubits (purple spheres), X-syndrome qubits (yellow) for phase parity detection of surrounding code qubits, and Z-syndrome qubits (green) for bit parity detection of surrounding code qubits. The physical connectivity for superconducting qubits can be realised via coupling every qubit to two quantum bus resonators, shown as wavy blue diamonds in the MAP. The device studied in this work (false-colored optical micrograph in b) embodies two half-plaquettes of the SC as circled in a, and allows for independent and simultaneous detection of X and Z errors on two-code qubits, shaded purple in b and labelled Q₁ and Q₃. (c) The circuit to implement the half-plaquette operations encodes the bit (ZZ) and phase (XX) parities of the two-code qubits' Bell state onto the respective syndrome qubits, Q₂ (green) and Q₄ (yellow). Arbitrary errors ɛ are intentionally introduced on the code qubit Q₁ and detected from the correlated measurement of the syndrome qubits. Q₂ (Q₄) is initialized to . A Hadamard operation, H, is applied to Q₄ before measurement.

Figure 2. Correlated syndrome single-shot histograms and quantum state tomography of code qubits.

The quantum state of the syndrome qubits reveals the entangled state of the code qubits. The colormaps show the single-shot histograms of the syndrome measurements on Q₂ and Q₄. The dashed white lines indicate the threshold used to condition the reconstruction of the code qubit states, represented by a Pauli vector. The pink-, blue- and purple-shaded regions signify Q₁, Q₃ and joint Pauli operators, respectively (black filled bars, experiment, white bars in background, ideal). Each of the possible four outcomes of correlated single-shot measurements of the syndrome qubits is mapped onto one of the four maximally entangled Bell states of the code qubits. Since we always prepare the code qubits in the codeword state at the beginning of the quantum process, when no error is applied to Q₁, state tomography of Q₁ and Q₃ conditioned on outcomes in the lower left quadrant {0,+} of the colormap recover the same state with fidelity 0.8491±0.0005 (a). Introducing an error ɛ equal to X (b), Z (c) and Y (d) on Q₁, and conditioning on outcomes in the upper left {1,+}, lower right {0,−} and upper right {1,−} quadrants results in the code qubits reconstructed as (fidelity 0.8195±0.0006), (with fidelity 0.8046±0.0005) and (fidelity 0.8148±0.0006), respectively. The X-syndrome qubit, Q₄, is found in its excited state when a phase-flip error has occurred (c,d), whereas the Z-syndrome qubit, Q₂, is found in its excited state as a result of bit-flip errors (b,d). The quoted uncertainties in reconstructed state fidelities are statistical (see Methods), but we note that systematic errors due to coherence time fluctuations, state preparation and measurement errors can lead to indifelity ∼0.01–0.02.

Figure 3. Syndrome qubits single-shot correlated measurement for different Y-error magnitudes.

The magnitude of each type of error in the code qubits can be extracted from the correlated single-shot traces of the syndrome qubits. Errors of ɛ=Y_θ, with θ∈[−π,π] are detected by both syndromes, as Y errors can be decomposed into a combination of bit- and phase-flip errors. As the magnitude of the Y error increases from 0 to π, the majority of the outcomes of the syndrome qubits changes from {M₂,M₄}={0,+} (black dots) to {M₂,M₄}={1,−} (blue dots), while the states {M₂,M₄}={1,+} (red dots) and {M₂,M₄}={0,−} (green dots), which indicate pure bit- and phase-flip errors, respectively, remain low probability. Solid lines are simple cosine fits to the data. Dashed lines are master-equation simulations that take into account the measured coherence times and assignment fidelities. Histograms of the correlated single-shot syndrome qubit measurements are shown in the density plots on top for θ∼{−π,−π/2,0,π/2,π}, as indicated by the vertical dashed lines, with the syndrome states corresponding to |θ|=π/2 showing significant populations in two quadrants.

Figure 4. Detection of arbitrary errors.

The probability of each type of error, identity (Id), X, Y or Z, is extracted from the correlated syndrome measurements for all the applied ɛ, as indicated above each panel. Dark blue bars represent the ideal outcome for each ɛ and teal bars are measurements calibrated by the full X, Y and Z error rotation curves. The errors labelled as R and H correspond to a Y_π/2X_π/2 operation, which maps the x–y–z axes in the Bloch sphere to y–z–x, and the Hadamard gate, respectively. The results are consistent with a higher uncertainty in the phase-flip error detection, likely due to decoherence during the full sequence and the order of syndrome detection.

Figure 5. Two-qubit randomized benchmarking.

Average population of the ground state of the target qubit, P₀, versus number of two-qubit Cliffords generated via ECR gates between (a) Q₁ and Q₂, (b) Q₂ and Q₃, (c) Q₃ and Q₄, and (d) Q₄ and Q₁. Each RB experiment is averaged over 50 different sequences. Fits to the experiments are shown as solid lines and yield average errors per two-qubit Clifford of (a) 0.0604±0.0006, (b) 0.0631±0.0007, (c) 0.0569±0.0015 and (d) 0.0353±0.0015. Inset shows ZX oscillations of the target qubit state population as a function of the cross-resonance drive length when the control qubit is in the ground (blue) or in the excited (red) state.

Figure 6. Experimental setup.

Detailed wiring scheme for all room temperature control electronics and internal configuration of the Oxford Instruments Triton dilution refrigerator.

Figure 7. Device and circuit schematic and qubit geometry.

The optical image shows all components of the device, including the four qubits, Q₁–Q₄, the four readout resonators R₁–R₄ and the four coupling buses B₁₂, B₂₃, B₃₄ and B₄₁. The readout resonators also serve as qubit control lines, with single- and two-qubit gates applied at frequencies ω_i with i∈{1,2,3,4}. Readout is performed at the resonator frequency ω_Mi. Each readout signal is reflected off a JPA, pumped at frequency ω_Pi, before being sent to a HEMT amplifier at 4 K. A blowup of one of the qubits is also shown, depicting the capacitor geometry as well as the coupling lines to the readout resonator (green coupler) and to the buses (red couplers). The black scale bar represents a length of 100 μm.

Figure 8. Quantum circuit and CNOT gate decomposition.

The ZZ and XX parity checks are performed on a pair of maximally entangled qubits. This entanglement is achieved in our architecture with one CNOT and one SWAP gate (a). The three CNOTs that define the SWAP gate can be combined with the following ZZ parity check operation to simplify the circuit and three CNOT gates can be eliminated (b). The final circuit implemented in our experiments has a total of five CNOT gates (c). We implement our CNOT gates using a simplified version of the gate, ECR^ij, consisting of two cross-resonance pulses of different sign separated by a π rotation in the control qubit. With that definition, a CNOT gate can be obtained with four single-qubit rotations plus a ECR^ij operation. An example, not unique, of such decomposition is shown in d. The complete gate sequence in our error detection experiments is presented in e, where the dark boxes indicate refocus pulses during every two-qubit gate on the two qubits not involved on it.

Figure 9. Continuous tracking of pure bit- and phase-flip errors.

Errors of ɛ=X_θ (a) and ɛ=Z_θ (b) with θ∈[−π,π] are applied to the code qubit Q₁. The syndrome qubit states {M₂,M₄}={0,+} (black), {M₂,M₄}={0,−} (green), {M₂,M₄}={1,+} (red) and {M₂,M₄}={1,−} (blue) indicate the magnitude and nature of the error ɛ. Since the ZZ and XX parities are encoded into Q₂ and Q₄, respectively, pure bit-flip errors are detected by Q₂, whereas pure phase-flip errors are detected by Q₄.