Figure 1

(a) Two-dimensional array implementation of the surface code. Filled circles represent data qubits and open circles represent measurement qubits. Stars in green (dark) represent $Z$ stabilizers and stars in yellow (light) represent $X$ stabilizers. The logical qubit is composed of a double defect in $Z$ stablizers. The logical Pauli operators for the logical qubit $Q$ are indicated in red. ${\widehat{Z}}_L$ is the product of Pauli-$Z$ operators of 8 data qubits on the ring in red and ${\widehat{X}}_L$ is the product of Pauli-$X$ operators of 7 data qubits on the chain in red. The code distance of this configuration is 7 and all logical qubits are separated with this code distance. Four logical qubits are shown in the figure. The sides of this two-dimensional array extend outwards where more logical qubits are located. (b) Representation of $Z$ stabilizer ${\widehat{Z}}_{1234}={\widehat{Z}}_1{\widehat{Z}}_2{\widehat{Z}}_3{\widehat{Z}}_4$ in the surface code (left) and quantum circuit (right) for one surface code cycle for measurement of ${\widehat{Z}}_{1234}$. The $Z$ stabilizer is indicated by a star in green. Data qubits 1, 2, 3, and 4 are located at terminals of the star and the measure qubit 0 is located at the center of the star. After initialization of qubit 0, stabilizer measurement includes four cnot operations followed by a projective measurement of qubit 0. (c) Representation of $X$ stabilizer ${\widehat{X}}_{1234}={\widehat{X}}_1{\widehat{X}}_2{\widehat{X}}_3{\widehat{X}}_4$ in the surface code (left) and quantum circuit (right) for one surface code cycle for measurement of ${\widehat{X}}_{1234}$. The $X$ stabilizer is indicated by a star in yellow. Data qubits 1, 2, 3, and 4 are located at terminals of the star and the measure qubit 0 is located in the center of the star. After initialization of qubit 0, the stabilizer measurement includes two Hadamard operations, four cnot operations, and a projective measurement of qubit 0.Reuse & Permissions

Figure 2

Implementation to perform a cnot gate between two smooth qubits [36]. The Control in, Target in, and ancilla $a_2$ are all smooth qubits. Ancilla $a_1$ is a rough qubit. Braiding operations are used to generate logical cnots between smooth and rough qubits. After three braiding operations, measurements $M_Z$ on $a_1$ and $M_X$ on Target in are performed. The measurement outcomes are used to interpret the output states.Reuse & Permissions

Figure 3

The logic circuit that implements the $S$ gate. The input state $|{\psi }_L\rangle$ is transformed to output state $S|{\psi }_L\rangle$.Reuse & Permissions

Figure 4

The logic circuit that implements the $T$ gate. The input state $|{\psi }_L\rangle$ is transformed to output state $T|{\psi }_L\rangle$.Reuse & Permissions

Figure 5

Distillation circuit for the $|A_L\rangle$ state [36]. Sixteen logical qubits are run through this circuit, yielding the output state $|{\psi }_L\rangle$ to be $|A_L\rangle$ with higher fidelity.Reuse & Permissions

Figure 6

Circuit for implementing the iterative phase estimation algorithm. Each qubit in the $N$-qubit input register corresponds to one of the $N$ spin-$\frac{1}{2}$ particles in the TIM model. Initially the $N$-qubit input quantum register is prepared with an estimate of the ground state of $H_I$. The output quantum register consists of a single-qubit recycled $M$ times. In each iteration, the output register gives the value of one bit in the $M$-bit binary expression of $\varphi =0.x_1...x_M$. The Hadamard gate, the controlled unitary gate $U\left(2^m\tau \right)$, and the single-qubit rotation gate $R_m$ are required. Details of the implementation can be found in Ref. [12].Reuse & Permissions

Figure 7

Circuit for the controlled-$U\left(2^m\tau \right)$ gate approximated using the second-order Trotter formula. The resulting decomposed gates are the controlled-$U_x\left(\theta \right)$ and controlled-$U_{zz}\left(\theta \right)$ gates. The controlled-$U\left(2^m\tau \right)$ gate corresponds to the $(M-m)\mathrm{th}$ measured bit in the binary fraction for the phase $\varphi$ and thus requires that the Trotter error must be less than the bit precision. Details of the implementation can be found in Ref. [12].Reuse & Permissions

Figure 8

The decomposition of the controlled-$U_x\left(\theta \right)$ into single-qubit $R_z$ gates and cnot gates. There is a single-control-bit multiple-target cnot gate in the circuit, which is different from the implementation in Ref. [12].Reuse & Permissions

Figure 9

The decomposition of the controlled-$U_{zz}\left(\theta \right)$ into single-qubit $R_z$ gates and cnot gates. There is a single-control-bit multiple-target cnot gate in the circuit, which is different from the implementation in Ref. [12].Reuse & Permissions

Figure 10

Numerical simulation of the surface code distance $d$ assuming a $N=100$ spin TIM problem, as a function of the desired maximum precision $M$.Reuse & Permissions

Figure 11

The limiting factor on the distillation speed of the $|A_L\rangle$ states. As the quantum simulation is proceeding, the distillation of $|A_L\rangle$ occurs simultaneously. Thus the speed of the distillation must be sufficient for the consumption of $|A_L\rangle$ states in the simulation. That is, during the time period of two-level distillation, the number of the distilled $|A_L\rangle$ states consumed in the simulation must be equal to that of the $|A_L\rangle$ states produced in the distillation. Here we consider the situation where $T$ gates are separated by only one $H$ gate and thus require the fastest distillation rate.Reuse & Permissions

Figure 12

Numerical simulations for the number of physical qubits, assuming a $N=100$ spin TIM problem, as a function of the desired maximum precision $M$.Reuse & Permissions

Figure 13

Numerical simulations of the number of logical cycles (upper panel) and the physical time (lower panel), assuming the $N=100$ spin TIM problem, as a function of the desired maximum precision $M$.Reuse & Permissions

Figure 14

The numerical calculations for the number of logical cycles $K$ required in the spin TIM simulation with the concatenation code as a function of the desired maximum precision $M$.Reuse & Permissions