Quantum Algorithms to Simulate Many-Body Physics of Correlated Fermions

Simulating strongly correlated fermionic systems is notoriously hard on classical computers. An alternative approach, as proposed by Feynman, is to use a quantum computer. We discuss simulating strongly correlated fermionic systems using near-term quantum devices. We focus specifically on two-dimensional (2D) or linear geometry with nearest-neighbor qubit-qubit couplings, typical for superconducting transmon qubit arrays. We improve an existing algorithm to prepare an arbitrary Slater determinant by exploiting a unitary symmetry. We also present a quantum algorithm to prepare an arbitrary fermionic Gaussian state with $O(N^2)$ gates and $O(N)$ circuit depth. Both algorithms are optimal in the sense that the numbers of parameters in the quantum circuits are equal to those describing the quantum states. Furthermore, we propose an algorithm to implement the 2D fermionic Fourier transformation on a 2D qubit array with only $O(N^{1.5})$ gates and $O(\sqrt{N})$ circuit depth, which is the minimum depth required for quantum information to travel across the qubit array. We also present methods to simulate each time step in the evolution of the 2D Fermi-Hubbard model—again on a 2D qubit array—with $O(N)$ gates and $O(\sqrt{N})$ circuit depth. Finally, we discuss how these algorithms can be used to determine the ground-state properties and phase diagrams of strongly correlated quantum systems using the Hubbard model as an example.

Figure 1

Quantum circuit for a Givens rotation on neighboring qubits: The part in the dotted box represents a rotation between the two states $|01⟩$ and $|10⟩$.

Figure 2

The fermionic spin orbitals on a $4\times 4$ array are mapped to qubits (the blue circles) on an array of the same dimensions using a snake-shaped JWT. The red arrow shows the direction of the JWT, and the qubits are numbered by their order in the JWT. Hopping terms between an odd-numbered row and the row below are called right-closed hopping terms, while those between an even-numbered row and the row below are called left-closed hopping terms.

Figure 3

The ancilla qubit $a$ is swapped with the system qubit $s$ to its left, and its state is then updated by the cnot gate.

Figure 4

The procedure to generate the desired parities for the right-closed hopping terms. The ancilla qubits (the orange ellipses) store the parity of the system qubits (the blue circles), where $j\text{−}k$ denotes the parity $(\sum _{l=j}^k{s}_l)$ mod 2. (a) The ancilla qubits are initially located at the right side of the lattice, with their states set to $|0⟩$. (b) They are then moved to the left while the parity stored in them being updated. The purple dashed lines represent cz gates between the ancilla and the system qubits for right-closed hopping terms. The cz gates in (a) can be omitted because the ancilla states are set to $|0⟩$ initially.

Figure 5

The cz gates involving one system qubit $s$ and two ancilla qubits $a$ and $b$. When $s\ne s^{\prime }$ and $a+b=a^{\prime }+b^{\prime }$ for the two basis states, a parity difference of $a+b$ is introduced. By comparison, no parity difference is introduced when both quantities are the same for the two basis states.

Figure 6

The procedure to generate the desired parities for the left-closed hopping terms. The ancilla qubits (the orange ellipses) store the parity of the system qubits (the blue circles), where $j\text{−}k$ denotes the parity $(\sum _{l=j}^k{s}_l)\mathrm{mod}2$. (a) Two cz gates (the dashed lines) are applied to the anilla qubit labeled 12 and the system qubits labeled 6 and 11. (b) Two cz gates are applied to the anilla qubit labeled 11-12 and the system qubits labeled 7 and 10.

Figure 7

The cz gates involving two system qubits $s$ and $t$ and one ancilla qubit $a$. When $a\ne a^{\prime }$ and $s+t=s^{\prime }+t^{\prime }$ for the two basis states, a parity difference of $s+t$ is introduced. By comparison, no parity difference is introduced when both quantities are the same.

Figure 8

The cz gates between system (blue) and ancilla (red) qubits. (a) Right-closed hopping terms (between an odd-numbered row and the row below it), e.g., the one involving qubit 2 in Fig. 4. (b) Left-closed hopping terms (between an even-numbered row and the row below it), e.g., the one involving qubits 6 and 11 in Fig. 6.

Figure 9

The cz gates between the system (blue) and ancilla (red) qubits in a single time step that introduce the desired parities to both the right-closed and left-closed hopping terms.

Figure 10

Quantum circuit to demonstrate the hopping terms between the second and third rows by rearranging the cz gates in Fig. 9.

Figure 11

The first and second parts in Fig. 9.

Figure 12

Parity basis of the columns. (a) The circuit maps the computational basis to the parity basis. (b) The parities stored in system qubit 10 (blue) and the corresponding ancilla (orange).

Figure 13

Equivalent circuits to Fig. 11. (a) The ancilla qubits are in the parity basis and the system qubits are in the original basis. (b) Both the system and the ancilla qubits are in the parity basis.

Figure 14

Phase diagram of the 2D Fermi-Hubbard model. The critical temperature $T_c$ depends on the value of hole doping, AFM stands for antiferromagnet.

Figure 15

One way to map the two spin states to qubits on a 2D qubit array using the JWT.

Figure 16

Quantum circuit to implement the Bogoliubov transformation in the BCS state.

Figure 17

(a)–(d) Implementing the hopping terms on a $4\times 4$ qubit array. The red arrow indicates the direction of the JWT and the system qubits (the blue circles) are numbered by their order in the JWT. The ancilla qubits (the orange ellipses) store the parity information of the system qubits, where $j\text{−}k$ denotes the parity $Z_{j\text{−}k}\equiv Z_j\cdots Z_k$. The purple dashed lines represent quantum gates between neighboring qubits.

Figure 18

Quantum circuit to implement the vertical hopping term between qubits 3 and 6.

Figure 19

A $2\times 9$ qubit ladder (two chains) is used to simulate the 2D Hubbard model on a $3\times 3$ lattice. The spin-up and -down orbitals are mapped to the upper and lower qubit chains, respectively, using two independent JWTs. The solid red lines represent nearest-neighbor interactions between qubits in the same JWT, and the dashed blue lines represent $ZZ$ interactions between qubits representing different spin states.

Figure 20

The in situ matrix transposition (e2) can be decomposed into elementary fermionic permutations involving only neighboring qubits (time goes from top to bottom). The first line is the initial ordering and the last line is the target ordering. In each line, the qubits plotted with the same bright color are involved in one elementary permutation, and their positions are updated in the next line. The gray qubits remain in the same place. The pink permutation can be implemented with three fswap gates involving only neighboring qubits, or by Hamiltonian evolution, as described in Appendix pp3. The rest of the permutations require a single fswap gate.

Figure 21

Fidelity with states from adiabatic evolution for various Trotter step numbers on a $2\times 2$ grid.

Figure 22

Fidelity with states from adiabatic evolution for various Trotter step numbers on a $3\times 2$ grid.

Figure 23

Fidelity with states from adiabatic evolution for various Trotter step numbers on a $4\times 2$ grid.

Figure 24

The cz gates between the system (blue) and ancilla (red) qubits in a single time step.

Figure 25

Rearranged circuit for hopping terms between the second and third rows.

Figure 26

Rearranged circuit for hopping terms between the third and fourth rows.

Figure 27

Rearranged circuit for hopping terms between the fourth and fifth rows.

Figure 28

Rearranged circuit for hopping terms between the fifth and sixth rows.