Variational quantum eigensolver with fewer qubits

We propose a qubit efficient scheme to study ground-state properties of quantum many-body systems on near-term noisy intermediate-scale quantum computers. One can obtain a tensor network representation of the ground state using a number of qubits smaller than the physical degrees of freedom. By increasing the number of qubits, one can exponentially increase the bond dimension of the tensor network variational ansatz on a quantum computer. Moreover, we construct circuits blocks which respect $\mathrm{U}\left(1\right)$ and $\mathrm{SU}\left(2\right)$ symmetries of the physical system and show that they can significantly speed up the training process and alleviate the gradient vanishing problem. To demonstrate the feasibility of the qubit efficient variational quantum eigensolver in a practical setting, we perform first-principles classical simulation of differentiable programming of the circuits. Using only six qubits, one can obtain the ground state of a $4\times 4$ square lattice frustrated Heisenberg model with fidelity over 97%. Arbitrarily-long-range correlations can also be measured on the same circuit after variational optimization.

Figure 1

Variational training of an MPS/PEPS prepared on a parametrized quantum circuit in the qubit efficient scheme. Shown at the top left is the quantum circuit with $R$ reusable qubits for the physical degrees of freedom and $V$ qubits for the virtual degrees of freedom. Green triangles represent the input single-qubit state initialized to $|0\rangle$ and the yellow square $q_k^{{\alpha }_k}$ is the $k\mathrm{th}$ output bit measured on the Pauli basis ${\alpha }_k$. After measurement, the first qubit is reset to $|0\rangle$ and then entangled with the remaining $V$ qubits before the next measurement. Each gray box represents a multilayer parametrized quantum circuit detailed in Fig. 2. Each block has a similar circuit structure and the circuit parameters ${\mathit{\theta }}_1,{\mathit{\theta }}_2,...,{\mathit{\theta }}_{N-V}$ are independent.

Figure 2

Internal structure of circuit blocks shown in Fig. 1. (a) General unstructured setup. Here $R_{{\theta }_i}^{\alpha }=e^{-i{\theta }_i{\sigma }_i\alpha /2}$ represents a parametrized single-qubit rotation gate. (b) The $\mathrm{U}\left(1\right)$ preserving block. The leftmost $X$ gate is applied only for odd steps to flip the input state to $|1\rangle$. The double crosses are the $swa{p}^{\alpha }$ gates [60]. (c) The $\mathrm{SU}\left(2\right)$ preserving block. The left and right panels are for odd and even steps, respectively. The last qubit is an ancilla for creating singlets between consecutive steps. The gates enclosed in the dashed box are repeated for $d$ times, where $d$ denotes the depth of the block.

Figure 3

Expanded view of the $\mathrm{SU}\left(2\right)$ symmetric quantum circuit ansatz shown in Fig. 2 for the $N=4\times 4$ Heisenberg model. The blocks inside a dashed box are repeated by $d$ times. In the qubit efficient scheme, the 1st–12th qubits are the same physical qubit which is reused after measurement. The 13th–16th qubits are the $V=4$ qubits which mediate entanglement in the final output state. The 17th qubit is the ancilla qubit for constructing the singlet product initial state. The operations for generating singlets (i.e., $X,H$, controlled gates, and swap gates) commute with parametrized swap gates in the dashed box, so that can be moved to the beginning of the circuit.

Figure 4

(a) Variational energy and (b) fidelity with respect to the exact ground state of a $4\times 4$ frustrated Heisenberg model as a function of gradient descent steps. The dashed line is the exact ground-state energy. The circuit consists of $V+1=5$ qubits. For the $\mathrm{SU}\left(2\right)$ ansatz there is one additional ancilla qubit [see Fig. 2].

Figure 5

Spin-spin correlation $\langle {\sigma }_i^z{\sigma }_j^z\rangle$ for (a) the frustrated Heisenberg model with $J_2=0.5$ and (b) the unfrustrated Heisenberg model with $J_2=0$. Here the ground state is obtained by using the $\mathrm{SU}\left(2\right)$ symmetric circuit shown in Fig. 2.

Figure 6

Variance of the gradient as a function of (a) system size $N$ and (b) number of virtual qubits $V$ for random initialized circuit parameters of structures shown in Fig. 2. The Hamiltonian is an open Heisenberg chain. The dashed lines are linear fits; however, in (b) we only use $V\le 6$ for fitting.

Figure 7

A projected pair of states inspired a variational quantum circuit ansatz with a $4\times 4$ square lattice layout. Yellow squares with the symbol $⥀$ are measure (on basis $\alpha =X,Y,Z)$ and reset operations. The other yellow squares are measurements without reset.

Figure 8

Energy as a function of training steps, with training parameters listed in Table 1.

Figure 9

(a) Circuits for preparing and measuring on a five-qubit cluster state. (b) Preparing and measuring the same state in the qubit efficient scheme using only two qubits.