[Paper Review] Approximate Quantum Compiling for Quantum Simulation: A Tensor Network based approach

Niall F. Robertson, Albert Akhriev+2 more|arXiv (Cornell University)|Jan 20, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces AQCtensor, a tensor network-based algorithm that uses approximate quantum compiling (AQC) to generate short-depth quantum circuits from Matrix Product States (MPS) for simulating time-evolved quantum many-body states. By optimizing all circuit parameters globally and employing a brickwork Ansatz with smart initialization, AQCtensor achieves constant-depth circuits for fixed simulation times, reducing circuit depth by at least an order of magnitude compared to existing MPS-to-circuit methods on 100-qubit systems.

ABSTRACT

We introduce AQCtensor, a novel algorithm to produce short-depth quantum circuits from Matrix Product States (MPS). Our approach is specifically tailored to the preparation of quantum states generated from the time evolution of quantum many-body Hamiltonians. This tailored approach has two clear advantages over previous algorithms that were designed to map a generic MPS to a quantum circuit. First, we optimize all parameters of a parametric circuit at once using Approximate Quantum Compiling (AQC) - this is to be contrasted with other approaches based on locally optimizing a subset of circuit parameters and "sweeping" across the system. We introduce an optimization scheme to avoid the so-called ``orthogonality catastrophe" - i.e. the fact that the fidelity of two arbitrary quantum states decays exponentially with the number of qubits - that would otherwise render a global optimization of the circuit impractical. Second, the depth of our parametric circuit is constant in the number of qubits for a fixed simulation time and fixed error tolerance. This is to be contrasted with the linear circuit Ansatz used in generic algorithms whose depth scales linearly in the number of qubits. For simulation problems on 100 qubits, we show that AQCtensor thus achieves at least an order of magnitude reduction in the depth of the resulting optimized circuit, as compared with the best generic MPS to quantum circuit algorithms. We demonstrate our approach on simulation problems on Heisenberg-like Hamiltonians on up to 100 qubits and find optimized quantum circuits that have significantly reduced depth as compared to standard Trotterized circuits.

Motivation & Objective

• To develop an efficient method for mapping time-evolved Matrix Product States (MPS) to short-depth quantum circuits for quantum simulation.
• To overcome the exponential fidelity decay (orthogonality catastrophe) that hinders global optimization of parametric circuits in large systems.
• To achieve constant circuit depth with respect to qubit count for fixed simulation time and error tolerance, unlike linear-depth staircase circuits.
• To outperform existing MPS-to-circuit algorithms in circuit depth and expressivity for simulating local Hamiltonian evolution.

Proposed method

• Uses a brickwork parametric quantum circuit structure with CNOT blocks to enable high expressivity without increasing depth.
• Applies Approximate Quantum Compiling (AQC) to globally optimize all circuit parameters simultaneously, avoiding local optimization pitfalls.
• Employs a 'Trotter-initialisation' scheme where initial parameters match the second-order Trotterized circuit, ensuring a favorable starting point for optimization.
• Utilizes a truncated Hilbert-Schmidt local cost function with bit-flip terms to mitigate gradient vanishing and improve optimization convergence.
• Employs a heuristic cost function C(k)L with k=1 to balance gradient magnitude and computational cost.
• Leverages classical tensor network methods (e.g., TEBD, TDVP) to classically prepare the target MPS before quantum circuit optimization.

Experimental results

Research questions

• RQ1 Can global optimization of parametric quantum circuits avoid the orthogonality catastrophe and enable efficient compilation of large-scale MPS?
• RQ2 Does a brickwork Ansatz with constant depth yield significantly shorter circuits than linear-staircase Ansätze for time-evolved MPS?
• RQ3 Can AQCtensor reduce circuit depth by at least an order of magnitude compared to existing MPS-to-circuit algorithms on 100-qubit systems?
• RQ4 How does the choice of cost function affect optimization convergence for large-scale quantum circuits?
• RQ5 Can the combination of classical MPS preparation and AQC-based circuit optimization enable simulation beyond classically simulable regimes?

Key findings

• For 100-qubit simulations of Heisenberg-like Hamiltonians, AQCtensor reduces circuit depth by at least an order of magnitude compared to the best existing generic MPS-to-circuit algorithms.
• The optimized circuits produced by AQCtensor achieve high fidelity (close to 1.0) with the target time-evolved states, as demonstrated on both next-to-nearest-neighbor and decorated hexagonal lattice Hamiltonians.
• The use of a truncated local cost function with bit-flip terms significantly improves gradient magnitude, enabling faster convergence during optimization.
• The circuit depth remains constant with increasing qubit count for fixed simulation time and error tolerance, in contrast to linearly scaling staircase circuits.
• The smart initialization from the Trotter circuit ensures the optimization starts from a physically relevant state, accelerating convergence.
• The method enables simulation of quantum many-body systems beyond classical simulatability by combining classical MPS preparation with optimized short-depth quantum circuits.