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arXiv:2310.00121 · Physical Review A

Tridiagonal matrix decomposition for Hamiltonian simulation on a quantum computer

Boris Arseniev, Dmitry Guskov, Richik Sengupta, Jacob Biamonte, Igor Zacharov

Published May 29, 2024• 4 citations
arXivHTMLPDF

Abstract

The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix ${2}^{n}\ifmmode\times\else\texttimes\fi{}{2}^{n}$ by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets $O(n)$. The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as a function of the number of qubits is lower than the oracle-based approach for $n<15$ and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.

DOI
10.1103/physreva.109.052629

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The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix 2n\ifmmode×\else\texttimes\fi2n{2}^{n}\ifmmode\times\else\texttimes\fi{}{2}^{n}2n\ifmmode×\else\texttimes\fi2n by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets O(n)O(n)O(n). The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as a function of the number of qubits is lower than the oracle-based approach for n<15n<15n<15 and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.
Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitioning sets of Paulis into commuting subsets, and show that the proposed approach can help to significantly reduce both the number of C N O T operations and circuit depth for Hamiltonians arising in quantum chemistry. The algorithms for simultaneous diagonalization are also applicable in the context of stabilizer states; in particular we provide novel four- and five-stage representations, each containing only a single stage of conditional gates.
To process a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operators, or gates. These operators are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BEing). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hamiltonian (LCH) or a Linear Combination of nitaries (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. In this work, we propose Hamiltonian matrices used to map the problem of interest, and that behave like a single qubit when expressed in the appropriate basis. It leads to a framework, the Special Tripotent Hamiltonian (STH) Framework, able to implement most of the typical problems considered for quantum computing. These methods address many problems implemented on QPUs, ranging from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), sparse matrices, and higher-order optimization problems. This work underlines interesting properties associated with the STH basic gate decomposition. These include the ability to switch between LCH and LCU, map non-Hermitian problems, and construct the quantum circuit queries required for quantum computing. We also provide a list of STH that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.
The simulation of multidimensional wave propagation with variable material parameters is a computationally intensive task, with applications from seismology to electromagnetics. While quantum computers offer a promising path forward, their algorithms are often analyzed in the abstract oracle model, which can mask the high gate-level complexity of implementing those oracles. We present a framework for constructing a quantum algorithm for the multidimensional wave equation with a variable speed profile. The core of our method is a decomposition of the system Hamiltonian into sets of mutually commuting Pauli strings, paired with a dedicated diagonalization procedure that uses Clifford gates to minimize simulation cost. Within this framework, we derive explicit bounds on the number of quantum gates required for Trotter–Suzuki-based simulation. Our analysis reveals significant computational savings for structured block-model speed profiles compared to general cases. Numerical experiments in three dimensions confirm the practical viability and performance of our approach. Beyond providing a concrete, gate-level algorithm for an important class of wave problems, the techniques introduced here for Hamiltonian decomposition and diagonalization enrich the general toolbox of quantum simulation.
We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for decomposing ddd-band diagonal matrices into Pauli strings that are grouped into mutually commuting sets. Using numerical simulations of the one-dimensional wave equation, we show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case, although they do not decrease the number of Trotter steps needed to preserve solution accuracy. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.
The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics—with the result encoded in the off-diagonal blocks of a Hamiltonian—using Hamiltonian evolutions of input operators. We show how to maintain this Hamiltonian block encoding , so that matrix operations can be composed one after another, and the entire quantum computation takes ≤ 2 ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadamard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain of interest, while the constructed polynomial is upper bounded by the target function over the entire unit interval. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the doubly factorized tensor hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with 1 ancilla qubit.
Constructing appropriate unitary matrix operators for new quantum algorithms and finding the minimum cost gate sequences for the implementation of these unitary operators is of fundamental importance in the field of quantum information and quantum computation. Evolution of quantum circuits faces two major challenges: complex and huge search space and the high costs of simulating quantum circuits on classical computers. Here, we use the group leaders optimization algorithm to decompose a given unitary matrix into a proper-minimum cost quantum gate sequence. We test the method on the known decompositions of Toffoli gate, the amplification step of the Grover search algorithm, the quantum Fourier transform, and the sender part of the quantum teleportation. Using this procedure, we present the circuit designs for the simulation of the unitary propagators of the Hamiltonians for the hydrogen and the water molecules. The approach is general and can be applied to generate the sequence of quantum gates for larger molecular systems.