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Preparation Circuits for Matrix Product States by Classical Variational Disentanglement

Refik Mansuroglu, Norbert Schuch

TL;DR

This work introduces Classical Variational Disentanglement (CVD), a classical method to compile matrix product states (MPS) into quantum circuits by learning a local-unitary disentangler $U=igl(\prod_i G_{i,i+1}\bigr)$ that collapses the MPS to a product state, after which the circuit inverse $U^\

Abstract

We study the classical compilation of quantum circuits for the preparation of matrix product states (MPS), which are quantum states of low entanglement with an efficient classical description. Our algorithm represents a near-term alternative to previous sequential approaches by reverse application of a disentangler, which can be found by minimizing bipartite entanglement measures after the application of a layer of parameterized disentangling gates. Since a successful disentangler is expected to decrease the bond dimension on average, such a layer-by-layer optimization remains classically efficient even for deep circuits. Additionally, as the Schmidt coefficients of all bonds are locally accessible through the canonical $Γ$-$Λ$ form of an MPS, the optimization algorithm can be heavily parallelized. We discuss guarantees and limitations to trainability and show numerical results for ground states of one-dimensional, local Hamiltonians as well as artificially spread out entanglement among multiple qubits using error correcting codes.

Preparation Circuits for Matrix Product States by Classical Variational Disentanglement

TL;DR

This work introduces Classical Variational Disentanglement (CVD), a classical method to compile matrix product states (MPS) into quantum circuits by learning a local-unitary disentangler that collapses the MPS to a product state, after which the circuit inverse $U^\

Abstract

We study the classical compilation of quantum circuits for the preparation of matrix product states (MPS), which are quantum states of low entanglement with an efficient classical description. Our algorithm represents a near-term alternative to previous sequential approaches by reverse application of a disentangler, which can be found by minimizing bipartite entanglement measures after the application of a layer of parameterized disentangling gates. Since a successful disentangler is expected to decrease the bond dimension on average, such a layer-by-layer optimization remains classically efficient even for deep circuits. Additionally, as the Schmidt coefficients of all bonds are locally accessible through the canonical - form of an MPS, the optimization algorithm can be heavily parallelized. We discuss guarantees and limitations to trainability and show numerical results for ground states of one-dimensional, local Hamiltonians as well as artificially spread out entanglement among multiple qubits using error correcting codes.
Paper Structure (16 sections, 3 theorems, 31 equations, 6 figures, 1 table)

This paper contains 16 sections, 3 theorems, 31 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classical Variational Disentanglement
  3. Overview
  4. Disentanglement by Entropy Minimization
  5. Classical Efficiency
  6. Absence of Barren Plateaus
  7. Numerical Experiments
  8. Spin Model Ground States
  9. 1D Fermi-Hubbard Ground State
  10. Entangled Logical Qubits
  11. Conclusion
  12. Classical Efficiency of CVD
  13. Bounds on Bond Dimension -- Proof of Lemma \ref{['lem:bond_dim']}
  14. Approximation Error Bound -- Proof of Lemma \ref{['lem:error']}
  15. Absence of Barren Plateaus in CVD -- Proof of Lemma \ref{['lem:barren']}
  16. ...and 1 more sections

Key Result

Lemma 1

Let $S_{\mathcal{A}, \alpha} (p_i) = \frac{1}{1-\alpha} \log\left( \sum_i p_i^{\alpha} \right)$ with $p_i := \lambda_i^2$ be the $\alpha$-Rényi entropy with $0 < \alpha <1$ given by the $\Lambda$ matrix separating an MPS into left- and right-subsystems $\mathcal{A}$ and $\mathcal{A}^c$. Let $p = \su

Figures (6)

  • Figure 1: Graphical representation of a unitary disentangler $U$ acting on an MPS $\vert \psi \rangle$. $U$ is built from a product of local gates $G_{i,i+1}$ (blue) acting on neighboring sites. When the bond dimension is reduced to one, a last layer of single-site gates can be read off to finally map the MPS to the computational basis state $\vert 0 \rangle^{\otimes n}$.
  • Figure 2: Bond dimensions for disentanglement of the ground state of a 1D Ising Model with skew magnetic field of strengths $h_x = 0.5$ and $h_z = 0.05$, see Eq. \ref{['eq:TFIM']}. After the target MPS is prepared with a DMRG, CVD is carried out using different values for the Rényi index $\alpha$ showcasing a suppression of the maximal bond dimension $D$ for small indices $\alpha$ in the presence of an error threshold $p = 10^{-7}$. All three algorithms converge to a per-site overlap error of approximately $\varepsilon = 10^{-4}$.
  • Figure 3: Disentanglement of the ground states of the Ising model with skew magnetic field $h_x=0.5, h_z=0.05$, the XY Model with $g=0$ and a magnetic field in X-direction, $h_x = 0.1$, and the Heisenberg XXZ Model with $J_z = 0.5$ and a skew magnetic field $h_z = 0.1$ and $h_x = 0.1$ defined on 50 sites. The target state has been determined using a DMRG routine and subsequently disentangled using a Rényi-index $\alpha = \frac{1}{2}$ on every third layer and $\alpha = 1$ else. The color plots (a) show a gradual decrease of the $\infty$-Rényi entropy starting from the boundary. As an approximation to the tail weight, it measures the distance of the MPS to a product state across the bond. Since the overlap error (b) also saturates for all three spin models, this indicates that there exist local minima with unresolved correlations. The system size scaling (c) of the final errors after 32 layers saturate at around $10^{-4}$ with one outlier for the 50-site Ising model.
  • Figure 4: Disentanglement of an MPS approximation to the ground state of the Fermi-Hubbard model with hopping parameter $t=1$ and Coulomb interaction strength $U=2$ on 25 electronic sites at half filling (12 spins up and 12 spins down). These are encoded into 50 qubits with each qubit the occupation of a spin-up or spin-down particle, respectively. Next to a plot of the tail weight (a), the energy of the prepared low-rank state is plotted against the layer count (b). The ground state energy is approximated by a DMRG calculation with bond dimension $D=1000$.
  • Figure 5: Disentanglement of logical Bell pair in a [5,1,3] and [11,1,5] stabilizer code in both the separated and the interlaced version as depicted in (d). In (a) and (b), the tail weights on each bond and for each layer in the disentangling process are shown and in (c), the overlap errors from backwards preparation of the initial MPS in the [11,1,5] code are shown. We find that in all cases a linear circuit depth is required to fully disentangle the state. Although the bond dimension is decreasing along the way, the overlap error undergoes an abrupt jump.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma 3