Quantum Simulation with Hybrid Tensor Networks

Tensor network theory and quantum simulation are, respectively, the key classical and quantum computing methods in understanding quantum many-body physics. Here, we introduce the framework of hybrid tensor networks with building blocks consisting of measurable quantum states and classically contractable tensors, inheriting both their distinct features in efficient representation of many-body wave functions. With the example of hybrid tree tensor networks, we demonstrate efficient quantum simulation using a quantum computer whose size is significantly smaller than the one of the target system. We numerically benchmark our method for finding the ground state of 1D and 2D spin systems of up to $8\times 8$ and $9\times 8$ qubits with operations only acting on $8+1$ and $9+1$ qubits, respectively. Our approach sheds light on simulation of large practical problems with intermediate-scale quantum computers, with potential applications in chemistry, quantum many-body physics, quantum field theory, and quantum gravity thought experiments.

Figure 1

Hybrid tensors and tensor contraction. (a) A low-rank classical tensor. (b) An $n+1$ rank tensor with $n$ indices representing an $n$-partite quantum system and 1 classical index. Each state could be prepared with (b1) different unitary as $|{\psi }^i⟩=U^i|\overline{0}⟩$ or (b2) different initial states as $|{\psi }^i⟩=U|{\overline{0}}^i⟩$. (c),(d) Tensor contraction between a quantum and a classical tensor. The contracted index could be (c) classical or (d) quantum, which shares the same mathematical definition, but is contracted in different ways. (e) Expectation values of local observables for a rank-($n+1$) tensor as a Hermitian observable $M^{i^{\prime },i}=⟨{\psi }^{i^{\prime }}|O_1\otimes O_2\otimes \cdots \otimes O_n|{\psi }^i⟩$. (e1) Suppose the index $i$ is classical and $|{\psi }^i⟩=U^i|\overline{0}⟩$, we get each $M^{i^{\prime },i}$ by measuring the ancillary qubit in the three Pauli bases and the other $n$ qubits in the $Z$ basis, with $U_1=U^i$, $U_2=U^{i^{\prime }}(U^i{)}^{†}$, and $U_M$ being the unitary that rotates to the observable basis. (e2) Suppose the index $i$ is classical and $|{\psi }^i⟩=U|{\overline{0}}^i⟩$, we use $U_1$ to prepare four input states $|{\overline{0}}^i⟩,|{\overline{0}}^{i^{\prime }}⟩,(|{\overline{0}}^i⟩+|{\overline{0}}^{i^{\prime }}⟩)/\sqrt{2},(|{\overline{0}}^i⟩+i|{\overline{0}}^{i^{\prime }}⟩)/\sqrt{2}$, and each $M^{i^{\prime },i}$ corresponds to a linear combination of the measurement results. (e3) Suppose the index $i$ is quantum, after applying the unitary $U$ for preparing the state $|\psi ⟩=U|\overline{0}⟩$; we measure $n$ qubits in the $Z$ basis and the other qubit in the Pauli $X$, $Y$, and $Z$ bases.

Figure 2

Hybrid tree tensor networks. (a) An example tree structure. (b) Connection of a quantum tensor and a classical tensor. (c) Connection of a quantum tensor and a classical TN. (d) Connection of two quantum tensors via a classical tensor. (e) Generalization of (d) with multiple subsystems. (f) Using classical tensors to represent local correlation and a quantum tensor to represent correlations between subsystems. (g) A quantum-quantum network. (h1)–(h4) An example for calculating expectation values of (g). To calculate (h1) the expectation value of local observables ${\otimes }_{i=1}^k{\otimes }_{j=1}^n{O}_j^i$, we first calculate (h2) the observable $M_s^{i_s^{\prime },i_s}$ for each tensor on the second layer with quantum circuits shown in Figs. 1 and 1(e2), which converts to the contraction in (h3) and the measurement using the quantum circuit (h4).

Figure 3

Numerical simulation for 1D and 2D quantum systems with hybrid TTN. (a) 1D spin cluster and 2D spin lattice with interactions (thin lines) on the boundary. The interactions of subsystems are represented by thick lines. We group 8 adjacent qubits and $3\times 3$ qubits on a square sublattice as subsystems for the 1D and 2D systems, respectively. (b) The ansatz circuit for the quantum tensors in Fig. 2. The circuits of both layers share similar structures with $d$ repetitions of circuits in the dashed box. Here, $R_{\alpha }$ ($\alpha \in \{\widehat{X},\widehat{Y},\widehat{Z}\}$) represents single-qubit rotation around $\alpha$ axis and the two-qubit gate is $R_{ZZ}({\theta }_i)=e^{-i{\theta }_i\widehat{Z}\otimes \widehat{Z}}$. The rotation angle (parameter) for each gate is initialized from a small random value and updated in each variational cycle. The circuit depths for $V$ (first layer tensor) and $U$ (second layer tensor) are $d(V)=6$ and $d(U)=8$. The additional unitary $M$ is inserted at the first and $[d/2+1]$th block of the first layer (b1) and the second layer (b2). (c),(d) Simulation results of the ground state energy. For the 1D and 2D cases, we compare $E$ to the reference results $E_0=E_{\mathrm{MPS}}$ and $E_0=E_{\mathrm{PEPS}}$ obtained from a standard density matrix renormalization group with bond dimension $\kappa =32$ and from PEPS imaginary time evolution with bond dimension $\kappa =5$ and maximum allowed boundary bond dimension of $\tilde{\kappa }=64$ during the contraction. We use the relative error $1-E/E_0$ to characterize the accuracy. The red dashed line (1D) and blue dash-dotted line (2D) correspond to the energy using tensor products of the ground state of local subsystems. The cyan dot (1D) and blue triangle (2D) are results obtained with hybrid TNs. (c1),(d1) Convergence toward the ground state for the 1D $8\times 8$ and 2D $9\times 4$ systems with $\lambda =1$, respectively. (c2),(d2) Error versus different subsystem coupling strength $\lambda$ for the 1D $8\times 8$ and 2D $9\times 4$ systems, respectively. (c3),(d3) Errors with different numbers of local subsystems with $\lambda =1$, respectively.