Surface code with decoherence: An analysis of three superconducting architectures

We consider realistic, multiparameter error models and investigate the performance of the surface code for three possible fault-tolerant superconducting quantum computer architectures. We map amplitude and phase damping to a diagonal Pauli “depolarization” channel via the Pauli twirl approximation, and obtain the logical error rate as a function of the qubit $T_{1,2}$ and state preparation, gate, and readout errors. A numerical Monte Carlo simulation is performed to obtain the logical error rates, and a leading-order analytic formula is derived to estimate their behavior below threshold. Our results suggest that scalable fault-tolerant quantum computation should be possible with existing superconducting devices.

Figure 1

Layout of the distance-3 surface code considered here. Empty circles denote data qubits, green (light gray) filled circles denote $X$-type and blue (dark gray) filled circles denote $Z$-type syndrome qubits. The dashed lines denote tunable qubit-qubit coupling. We refer to this hardware design as the textbook architecture.

Figure 2

Schematic diagram of the distance-3 Helmer architecture. The circles represent superconducting qubits, with “idle” frequencies indicated by their colors (grayscales). The horizontal and vertical gray (magenta) rectangles are resonators. All horizontal (vertical) resonators have the same frequency.

Figure 3

Schematic diagram of the architecture discussed by DiVincenzo [26] for code distance $d=3$. The filled circles with boundaries represent qubits, the squares with boundaries represent resonators, and the colors (grayscales) of both denote their fixed frequencies. The unbounded circles are for the eye and indicate whether a given block is for data (dark gray), $X$-type syndrome (light green or light gray), or $Z$-type syndrome (blue or medium gray). A possible frequency allocation for all the components is shown.

Figure 4

A schematic diagram of distance-3 surface code is shown. Two possible error chains, $X_{\mathrm{L}}$ (purple and horizontal) and $Z_{\mathrm{L}}$ (magenta and vertical), are displayed and various terminologies used in this paper are illustrated. Syndrome $Z$ operators are shown in green (labeled by $Z$) and syndrome $X$ operators are in yellow (labeled by $X$). An error chain starting and ending at the same boundary is referred to as a “clasp” and is shown in dark gray color.

Figure 5

A schematic diagram of a surface code error correction cycle is shown. The dark gray leftmost region (red region) contains state preparation, the medium gray middle part (blue region) contains four consecutive cnot operations, and the light gray (green region) highlights the measurements of syndrome $Z$ and $X$ qubits.

Figure 6

Plot of analytic estimate of logical $X$ error probability per cycle vs. single physical qubit error probability per timestep. Solid lines denote numerical estimates via Monte Carlo simulation while dashed lines are obtained from our analytical formula given by Eq. (A6).

Figure 7

Logical $X$ and $Z$ error rate per cycle is shown as a function of coherence time $T_1$ for the textbook architecture. Plots for $d=3$ are shown in blue (dark gray) and those for $d=5$ are shown in red (light gray).

Figure 8

Logical $X$ and $Z$ error rate per cycle is shown as a function of coherence time $T_1$ for the Helmer architecture. Plots for $d=3$ are shown in blue (dark gray) and those for $d=5$ are shown in red (light gray).

Figure 9

Logical $X$ and $Z$ error rates per cycle are shown as a function of coherence time $T_1$ for the DiVincenzo architecture. Plots for $d=3$ are shown in blue (dark gray) and those for $d=5$ are shown in red (light gray).

Figure 10

Various possible fixed-coupling-based architectures are shown for $d=3$ surface code. The circles denote qubits, squares denote resonators, and various colors (grayscales) denote a possible frequency allocation. (a) An architecture where superconducting qubits are arranged in a two-dimensional square lattice each coupled to its nearest neighbor with fixed couplers. (b) An architecture where superconducting qubits are used for data qubits and resonators for syndrome qubits coupled via fixed couplers. Each resonator is also coupled to another qubit required for readout. (c) Same as architecture (b) except for the fact that each qubit is also coupled to another resonator used as its memory. (d) In this architecture each qubit in a two-dimensional square lattice is coupled to its nearest neighbor via a resonator.

Figure 11

Data qubits of a single row in a distance-5 surface code is shown with red (light gray) filled circles in two subsequent time slices. The blue (dark gray) filled circles denote measurement locations. An error in any measurement location generates two adjacent timelike syndrome events.

Figure 12

Plot of leakage probability from $\left|11\right.〉$ state under decoherence for various $g$ during a cz operation vs total cz operation time.