Shallow quantum circuits for efficient preparation of Slater determinants and correlated states on a quantum computer

Fermionic Ansatz state preparation is a critical subroutine in many quantum algorithms such as the variational quantum eigensolver for quantum chemistry and condensed-matter applications. The shallowest circuit depth needed to prepare Slater determinants and correlated states to date scales at least linearly with respect to the system size $N$. Inspired by data-loading circuits developed for quantum machine learning, we propose an alternate paradigm that provides shallower, yet scalable, $O\left(d{log}_2^{2}N\right)$ two-qubit gate-depth circuits to prepare such states with $d$ fermions, offering a subexponential reduction in $N$ over existing approaches in second quantization, enabling high-accuracy studies of $d\ll O(N/{log}_2^{2}N)$ fermionic systems with larger basis sets on near-term quantum devices.

Figure 1

Different approaches for preparing $d$-occupied Slater determinants of $N$ modes on a quantum computer, assuming Jordan-Wigner mapping. (a) Existing approaches use a linear-depth mesh of fermionic single-excitation gates to apply a fermionic basis transformation to a reference state [13, 14, 15]. (b) The proposed Clifford loaders via the Givens-rotation approach applies a sequence $\widehat{C}$ of gates $d$ times to an all-zero state of $N$ qubits ${|0\rangle }^{\otimes N}$. (c) $\widehat{C}$ consists of two products of multiple Givens rotations, $\widehat{D}$ and ${\widehat{D}}^{†}$, that sandwich a Pauli-$X$ gate on the first qubit. (d) The Givens rotations ${\widehat{U}}_{\mu \nu }$ are arranged in a binary tree. (e) ${\widehat{U}}_{\mu \nu }$ is decomposed using Pauli rotation gates $R_x$ and $R_z$ acting on qubit $\mu$, Hadamard gates $H$ acting on qubit $\nu$, and cnot ladders $B_{\mu \nu }$ acting on all qubits between $\mu$ and $\nu$. $\theta$ and $\vec{A}$ are scalar and vector parameters, respectively.

Figure 2

Linear-depth cascading cnot ladder $B$ in the decomposition of the Givens-rotation gate $\widehat{U}$ is replaced by a nonequivalent logarithmic-depth circuit that has no effect on $\widehat{U}$.

Figure 3

(a) A product of multiple Givens rotations ${\widehat{D}}_L^{†}$ that is composed of Givens-rotation gates ${\widehat{U}}_{\mu ,\nu }^{\left(L\right)}$ in a binary-tree arrangement on the quantum circuit. (b) ${\widehat{U}}_{\mu ,\nu }^{\left(L\right)}$ is decomposed using Pauli rotation gates $R_x$, acting on qubit $L\mu$, and $R_z$, acting on qubit $L(\mu -1)+1$, and Hadamard gates $H$ and cnot ladders $B_{\mu \nu }^{\left(L\right)}$ that act on all $2L-1$ qubits in $\left\{L\right(\mu -1)+1,...,L\mu -1\}$ and $\left\{L\right(\nu -1)+1,...,L\nu \}$, with $B_{\mu \nu }^{\left(L\right)}$ also acting on $\nu -\mu$ additional qubits in $\{L\mu ,L(\mu +1),L(\mu +2),...,L(\nu -1\left)\right\}$. $\theta$ and $\vec{G}$ are scalar and vector parameters, respectively.

Figure 4

Estimated two-qubit gate depth per occupied mode $d$ to prepare an $N$-mode Slater determinant $|{\mathrm{\Psi }}_1\rangle$ and an $L=2$ pairwise correlated Ansatz state $|{\mathrm{\Psi }}_2\rangle$ using Clifford loaders compared to existing $d$-independent linear-depth approaches.

Figure 5

Numerically calculated fraction of the electronic correlation energy ratio $E_{\text{pair}}/E_{\text{corr}}$ captured by an optimized pairwise correlated Ansatz state for hydrogen chains up to ${\text{H}}_6$ at a fixed interatomic distance of 1.4 bohrs for various basis-set mixtures up to 20 qubits.

Figure 6

The pairwise Givens-rotation gate ${\widehat{U}}_{\mu ,\nu }^{\left(2\right)}$ is decomposed using Pauli-rotation gates $R_x$, which acts on qubit $2\mu$ that correspond to ${\widehat{Y}}_{2\mu }$, and $R_z$, which acts on qubit $2\mu -1$, and Hadamard gates $H$ and cnot ladders $B_{\mu \nu }^{\left(L\right)}$ that act on all three qubits in $\{2\mu -1,2\nu -1,2\nu \}$ that correspond to ${\widehat{X}}_{2\mu -1}{\widehat{X}}_{2\nu -1}{\widehat{X}}_{2\nu }$, with $B_{\mu \nu }^{\left(L\right)}$ also acting on $\nu -\mu$ additional qubits in $\{2\mu ,2(\mu +1),2(\mu +2),...,2(\nu -1\left)\right\}$. $\theta$ is a scalar parameter.