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arXiv:2108.03282 · Physical Review A

Algebraic compression of quantum circuits for Hamiltonian evolution

Efekan Kökcü, Daan Camps, Lindsay Bassman Oftelie, J. K. Freericks, Wibe Albert de Jong, Roel Van Beeumen, Alexander F. Kemper

Published Mar 10, 2022• 46 citations
arXivHTMLPDF

Abstract

Unitary evolution under a time-dependent Hamiltonian is a key component of simulation on quantum hardware. Synthesizing the corresponding quantum circuit is typically done by breaking the evolution into small time steps, also known as Trotterization, which leads to circuits the depth of which scales with the number of steps. When the circuit elements are limited to a subset of $SU(4)$---or equivalently, when the Hamiltonian may be mapped onto free fermionic models---several identities exist that combine and simplify the circuit. Based on this, we present an algorithm that compresses the Trotter steps into a single block of quantum gates using algebraic relations between adjacent circuit elements. This results in a fixed depth time evolution for certain classes of Hamiltonians. We explicitly show how this algorithm works for several spin models, and demonstrate its use for adiabatic state preparation of the transverse field Ising model.

DOI
10.1103/physreva.105.032420

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Unitary evolution under a time-dependent Hamiltonian is a key component of simulation on quantum hardware. Synthesizing the corresponding quantum circuit is typically done by breaking the evolution into small time steps, also known as Trotterization, which leads to circuits the depth of which scales with the number of steps. When the circuit elements are limited to a subset of SU(4)SU(4)SU(4)---or equivalently, when the Hamiltonian may be mapped onto free fermionic models---several identities exist that combine and simplify the circuit. Based on this, we present an algorithm that compresses the Trotter steps into a single block of quantum gates using algebraic relations between adjacent circuit elements. This results in a fixed depth time evolution for certain classes of Hamiltonians. We explicitly show how this algorithm works for several spin models, and demonstrate its use for adiabatic state preparation of the transverse field Ising model.
In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.
Dynamic quantum simulation is a leading application for achieving quantum advantage. However, high circuit depths remain a limiting factor on near-term quantum hardware. We present a compilation algorithm based on Matrix Product Operators for generating compressed circuits enabling real-time simulation on digital quantum computers, that for a given depth are more accurate than all Trotterizations of the same depth. By the efficient use of environment tensors, the algorithm is scalable in depth far beyond prior work, and we present circuit compilations of up to 64 layers of SU(4) gates. Surpassing only 1D circuits, our approach can flexibly target a particular quasi-2D gate topology. We demonstrate this by compiling a 52-qubit 2D Transverse-Field Ising propagator onto the IBM Heavy-Hex topology. For all circuit depths and widths tested, we produce circuits with smaller errors than all equivalent depth Trotter unitaries, corresponding to reductions in error by up to 4 orders of magnitude and circuit depth compressions with a factor of over 6.
We describe tensor network algorithms to optimize quantum circuits for adiabatic quantum computing. To suppress diabatic transitions, we include counterdiabatic driving in the optimization and utilize variational matrix product operators to represent adiabatic gauge potentials. Traditionally, Trotter product formulas are used to turn adiabatic time evolution into quantum circuits and the addition of counterdiabatic driving increases the circuit depth per time step. Instead, we classically optimize a parameterized quantum circuit of fixed depth to simultaneously capture adiabatic evolution together with counterdiabatic driving over many time steps. The methods are applied to the ground-state preparation of quantum Ising chains with transverse and longitudinal fields. We show that the classically optimized circuits can significantly outperform Trotter product formulas. Additionally, we discuss how the approach can be used for combinatorial optimization. Published by the American Physical Society 2024
Quantum computing is a promising technology that harnesses the peculiarities of quantum mechanics to deliver computational speedups for some problems that are intractable to solve on a classical computer. Current generation noisy intermediate-scale quantum (NISQ) computers are severely limited in terms of chip size and error rates. Shallow quantum circuits with uncomplicated topologies are essential for successful applications in the NISQ era. Based on matrix analysis, we derive localized circuit transformations to efficiently compress quantum circuits for simulation of certain spin Hamiltonians known as free fermions. The depth of the compressed circuits is independent of simulation time and grows linearly with the number of spins. The proposed numerical circuit compression algorithm behaves backward stable and scales cubically in the number of spins enabling circuit synthesis beyond O(103)\mathcal{O}(10^3)O(103) spins. The resulting quantum circuits have a simple nearest-neighbor topology, which makes them ideally suited for NISQ devices.
Solving the electronic structure problem via unitary evolution of the electronic Hamiltonian is one of the promising applications of digital quantum computers. One of the practical strategies to implement the unitary evolution is via Trotterization, where a sequence of short-time evolutions of fast-forwardable (i.e. efficiently diagonalizable) Hamiltonian fragments is used. Given multiple choices of possible Hamiltonian decompositions to fast-forwardable fragments, the accuracy of the Hamiltonian evolution depends on the choice of the fragments. We assess efficiency of multiple Hamiltonian partitioning techniques using fermionic and qubit algebras for the Trotterization. Use of symmetries of the electronic Hamiltonian and its fragments significantly reduces the Trotter error. This reduction makes fermionic-based partitioning Trotter errors lower compared to those in qubit-based techniques. However, from the simulation-cost standpoint, fermionic methods tend to introduce quantum circuits with a greater number of T-gates at each Trotter step and thus are more computationally expensive compared to their qubit counterparts.
Digital quantum simulation relies on Trotterization to discretize time evolution into elementary quantum gates. On current quantum processors with notable gate imperfections, there is a critical trade-off between improved accuracy for finer time steps, and increased error rate on account of the larger circuit depth. We present an adaptive Trotterization algorithm to cope with time dependent Hamiltonians, where we propose a concept of piecewise "conserved" quantities to estimate errors in the time evolution between two (nearby) points in time; these allow us to bound the errors accumulated over the full simulation period. They reduce to standard conservation laws in the case of time independent Hamiltonians, for which we first developed an adaptive Trotterization scheme [H. Zhao et al., Making Trotterization adaptive and energy-self-correcting for NISQ devices and beyond, PRX Quantum 4, 030319 (2023).2691-339910.1103/PRXQuantum.4.030319]. We validate the algorithm for a time dependent quantum spin chain, demonstrating that it can outperform the conventional Trotter algorithm with a fixed step size at a controlled error.