Variational quantum algorithm for nonequilibrium steady states

We propose a quantum-classical hybrid algorithm to simulate the nonequilibrium steady state of an open quantum many-body system, named the dissipative-system variational quantum eigensolver (dVQE). To employ the variational optimization technique for a unitary quantum circuit, we map a mixed state into a pure state with a doubled number of qubits and design the unitary quantum circuit to fulfill the requirements for a density matrix. This allows us to define a cost function that consists of the time evolution generator of the quantum master equation. Evaluation of physical observables is, in turn, carried out by a quantum circuit with the original number of qubits. We demonstrate our dVQE scheme by both numerical simulation on a classical computer and actual quantum simulation that makes use of the device provided in Rigetti quantum cloud service.

Figure 1

Variational quantum circuit as the ansatz for the vector representation of the NESS of open quantum system with three physical qubits. Both the physical and ancillary qubits, denoted as ${|0\rangle }_{\mathcal{P}}$ and ${|0\rangle }_{\mathcal{A}}$, respectively, are initialized to zero. To impose the Hermiticity and positive-semidefiniteness that are required for density matrices, quantum gates for eigenvalue distribution $\tilde{D}$, CNOT gates between physical and ancillary qubits, and basis transformation $V$ operate sequentially.

Figure 2

Examples for the quantum circuit for eigenvalue distribution $\tilde{D}$. (a) The entangled type which consists of the repeating structure of single-qubit gates and controlled Y rotation. Here, $D_1$ denotes the repetition of the block in the bracket to enhance the representability of the ansatz. (b) The decoupled type with solely the single-qubit Y rotations.

Figure 3

Example for the quantum circuit for the basis transformation $V$. Here, we adopt the so-called hardware-efficient ansatz, which repeats the block consisting of an arbitrary single-qubit gate for each qubit and CZ gates between neighboring qubits. The repetition number of the block in the bracket is denoted as $D_2$.

Figure 4

(a) Quantum simulation and (b) noisy numerical simulation of magnetization curves (left axis) for the dissipative quantum Ising model. The colors, blue, orange, and green, correspond to $x$, $y$, and $z$ components of the mean magnetizations respectively for each dVQE (filled circles) and exact diagonalization (dotted lines). The unfilled triangles denote the infidelity $1-F$ (right axis) which is typically ${10}^{-2}$. The red arrows indicate the corresponding axes for data points. The dissipation amplitudes are taken as ${\gamma }_1=1$ and ${\gamma }_2=0.5$. The eigenvalue distribution ansatz is decoupled (entangled) types shown pictorially in Fig. 2 [Fig. 2] with the repetition numbers taken as $D_2=0$ ($D_1=D_2=1$) for system size $N=1\left(2\right)$ under quantum (numerical) simulation. Each Pauli term is sampled over 400 times.

Figure 5

Numerical simulation of the persistent current (left axis) and infidelity (right axis) for the reservoir-engineered coupled QED model. The blue filled circles and dashed line are the persistent current calculated by the dVQE scheme and exact diagonalization, respectively, and the unfilled triangles denote the infidelity. The red arrows indicate the corresponding axes for the data points. The dissipation amplitudes are taken as ${\gamma }_1=0.3$ and ${\gamma }_2=0.5$ for damping and dephasing, respectively. We employ the entangled type for the eigenvalue distribution ansatz, and take the repetition numbers as $D_1=D_2=2.$ Each Pauli term is sampled over 400 times.

Figure 6

Deviation between trace distance in the matrix representation and 2-norm in the vector representation. Here, 1000 independent data with varying range of noise is displayed. The blue, orange, and green dots denote data for random mixed states of $n=4,8,12$ qubits, and the black dotted line indicates $d_v=d_m$.

Figure 7

Numerical simulation for the landscape of the cost function with respect to single rotational angle for (a) the RY gate in $\tilde{D}$, (b) the controlled RY gate in $\tilde{D}$, and (c) the RY gate in $V$. The blue dots are the error mitigated value obtained by sampling from noisy quantum circuits, the black lines obtained from fitting into function with appropriate modes, and the red line is the exact value. The Liouvillian operator is identical to the one employed in Sec. 4b with the parameters given as $J=0.3,\lambda =0,\mu =1,{\gamma }_1=0.3,{\gamma }_2=0.5,\text{and}\theta =\pi /8$. We take entangled type for the eigenvalue distribution ansatz with the repetition numbers $D_1=D_2=1$. Each Pauli term is sampled over 400 times.

Figure 8

Quantum simulation of the cost function landscape with gate error mitigation. The blue, yellow, and red unfilled circles are results from a redundant quantum circuit with $\mathcal{E}=1,3,$ and 5, respectively. Error mitigated values obtained from linear extrapolation are shown as the purple filled circles, and the exact values are denoted by the real line. The cost function is defined from the Liouvillian operator identical to the one in Sec. 4a with the parameters given as $g=0.5,{\gamma }_1=1,{\gamma }_2=0.5$ with system size $N=1$. We use the decoupled type for the eigenvalue distribution ansatz, and the repetition number for the basis transformation is $D_2=0$. Here, the cost function is computed for the parameter in the eigenvalue distribution, and each Pauli term is sampled over 400 times.

Figure 9

Noiseless numerical simulation of magnetization curves (left axis) for the dissipative quantum Ising model. The style of the figure is identical to Fig. 4. Here, the dissipation amplitudes are taken as ${\gamma }_1=1$ and ${\gamma }_2=0.5$ for system size $N=8$. The eigenvalue distribution ansatz is the entangled type shown pictorially in Fig. 2. The repetition numbers are taken as $D_1=D_2=1$, and hence the circuit consists of 37 parameters in total.