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IEEE International Conference on Quantum Computing and Engineering (QCE)

Scalable Quantum Circuits of Exponential of Pauli-Strings and its Applications to Hamiltonian Simulations

Rohit Sarma Sarkar, Sabyasachi Chakraborty, Bibhas Adhikari

Published Aug 30, 2025• 1 citation
Publisher

Abstract

In this paper, we design quantum circuits for the exponential of scaled$n$-qubit Pauli-strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity and$X$gates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with the$n$-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Paulistring is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for ($n+1$)-qubit systems can be constructed from$n$-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include 2-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers, and observe that the noisy numerical simulation closely resembles the noiseless one when gate and idle errors are on the order of$O\left(10^{-3}\right)$or smaller.

DOI
10.1109/qce65121.2025.00201

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In this paper, we design quantum circuits for the exponential of scalednnn-qubit Pauli-strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity andXXXgates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with thennn-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Paulistring is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for (n+1n+1n+1)-qubit systems can be constructed fromnnn-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include 2-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers, and observe that the noisy numerical simulation closely resembles the noiseless one when gate and idle errors are on the order ofO(10−3)O\left(10^{-3}\right)O(10−3)or smaller.
In this paper, we design quantum circuits for the exponential of scaled nnn-qubit Pauli strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity and XXX gates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with the nnn-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Pauli-string is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for (n+1)(n+1)(n+1)-qubit systems can be constructed from nnn-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include 222-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers and observe that the noisy simulation closely resembles the noiseless one when gate and idle errors are on the order of O(10−3)O(10^{-3})O(10−3) or smaller.
In this paper, we develop a Lie group theoretic approach for parametric representation of unitary matrices. This leads to develop a quantum neural network framework for quantum circuit approximation of multi-qubit unitary gates. Layers of the neural networks are defined by product of exponential of certain elements of the Standard Recursive Block Basis, which we introduce as an alternative to Pauli string basis for matrix algebra of complex matrices of order 2n2^n2n. The recursive construction of the neural networks implies that the quantum circuit approximation is scalable i.e. quantum circuit for an (n+1)(n+1)(n+1)-qubit unitary can be constructed from the circuit of nnn-qubit system by adding a few CNOT gates and single-qubit gates.
Electronic Hamiltonian simulation is commonly formulated by mapping fermionic operators to qubit operators and subsequently expanding the resulting ladder-operator products into Pauli strings. While general, this procedure obscures the higher-level fermionic structure and can hide opportunities for circuit optimization. Building on ladder-string-pair (Lasp) diagonalization originally developed for Hamiltonian simulation of partial differential equations, we construct time-evolution circuits for the second-quantized electronic Hamiltonian without performing a Pauli expansion. For the most general case of complex-valued coefficients, we present a time-evolution circuit for a two-body fermionic Lasp operator that corresponds to 16 Pauli strings in the Pauli-expansion approach but does not suffer from Trotter error at this stage. Expanding the optimization scope from a single operator to a triad of three fermionic Lasp operators sharing the same four spin-orbital indices enables systematic cancellation of CX gates, reducing the CX-gate count from 36 to 12 in the example considered, still without introducing Trotter error at this stage. For nnn spin orbitals, further expanding the optimization scope to a sequence of O(n)O(n)O(n) suitably ordered triads enables a cascade of CX-gate reductions across triad boundaries, reducing the CX-gate count from O(n2)O(n^2)O(n2) to O(n)O(n)O(n). The Lasp-based approach also naturally accommodates controlled time evolution and yields further optimizations for real-valued Hamiltonians. These results demonstrate that the Lasp-based approach enables more efficient time-evolution circuits by preserving high-level circuit structures and thereby expanding the scope of optimization, providing a systematic route toward more efficient electronic Hamiltonian simulation.
Decomposing unitary operations into native gates is an essential step for implementing quantum algorithms. For qubit-based devices, where native gates are typically single- and two-qubit operations, a range of decomposition techniques have been developed. In particular, efficient algorithms exist for decomposing exponentials of Pauli strings while taking hardware topology in account. Motivated by the growing interest in qutrit-based quantum computing, we develop analogous decomposition methods for qutrit systems. Specifically, we introduce an algorithm that decomposes the exponential of an arbitrary tensor product of Weyl-Heisenberg operators (plus their Hermitian conjugation) into single- and two-qutrit gates. We further extend this approach to unitaries generated by Gell-Mann string (i.e., a tensor product of Gell-Mann matrices). Since both Gell-Mann matrices and Weyl-Heisenberg operators form (together with identity) complete operator bases of qutrit operators, we can use this result also to decompose any multi-qutrit gate that is diagonal up to single-qutrit rotations. As a practical application, we use our method to decompose the layers of the quantum approximate optimization algorithm for qutrit-based implementations of the graph k-coloring problem. For values of kkk well-suited to qutrit architectures (e.g., k=3k=3k=3 or in general k=3nk=3^nk=3n), our approach yields significantly shallower circuits compared to qubit-based implementations, an advantage that grows with problem size, while also requiring a smaller total Hilbert space dimension. Finally, we also address the routing challenge in qutrit architectures that arises due to the limited connectivity of the devices. In particular, we generalize the Steiner-Gauss method, originally developed to reduce CNOT counts in qubit circuit, to optimize gate routing in qutrit-based systems.
We introduce a general framework for constructing compact quantum circuits that implement the real-time evolution of Hamiltonians of the form H=σPBH = σP_BH=σPB​, where σσσ is a Pauli string commuting with a projection operator PBP_BPB​ onto a subspace of the computational basis. Such Hamiltonians frequently arise in quantum algorithms, including constrained mixers in QAOA, fermionic and excitation operators in VQE, and lattice gauge theory applications. Additionally, we construct transposition gates, widely used in quantum computing, that scale more efficiently than the best known constructions in literature. Our method emphasizes the minimization of non-transversal gates, particularly T-gates, critical for fault-tolerant quantum computing. We construct circuits requiring O(n∣B∣)\mathcal{O}(n|B|)O(n∣B∣) CX gates and O(n∣B∣+log⁡(∣B∣)log⁡(1/ε))\mathcal{O}(n |B| + \log(|B|) \log (1/ε))O(n∣B∣+log(∣B∣)log(1/ε)) T-gates, where nnn is the number of qubits, ∣B∣|B|∣B∣ the dimension of the projected subspace, and εεε the desired approximation precision. For subspaces that are generated by Pauli X-orbits we further reduce complexity to O(nlog⁡∣B∣)\mathcal{O}(n \log |B|)O(nlog∣B∣) CX gates and O(n+log⁡(1ε))\mathcal{O}(n+\log(\frac{1}ε))O(n+log(ε1​)) T gates. Our constructive proofs yield explicit algorithms and include several applications, such as improved transposition circuits, efficient implementations of fermionic excitations, and oracle operators for combinatorial optimization. In the sparse case, i.e. when ∣B∣|B|∣B∣ is small, the proposed algorithms scale favourably when compared to direct Pauli evolution.
Controlled time-evolution circuits select forward or backward Hamiltonian time evolution according to the state of an ancilla qubit. They are fundamental building blocks in quantum eigenvalue transformation of unitaries, Hamiltonian filtering, and related quantum algorithms. A direct realization adds ancilla control to the elementary gates of the time-evolution circuit and therefore increases the two-qubit gate count, compiled T depth and CX depth. We develop an efficient recursive algorithm that, for an arbitrary Pauli-sum Hamiltonian, partitions the input Pauli terms into groups and assigns to each group a sign-flip Pauli string that anti-commutes with the in-group terms, thereby removing ancilla control from the grouped time-evolution blocks. Numerical benchmarks on random Hamiltonians and structured spin Hamiltonians show reductions in compiled T depth and compiled CX depth. For a Kagome Hamiltonian with 24 spins under full connectivity, the proposed construction reduces the compiled T depth by 85.2% and the compiled CX depth by 68.9%, compared with a conventional implementation that decomposes the Hamiltonian into individual Pauli terms and implements the controlled time evolution of each term by directly adding the ancilla qubit to the corresponding Pauli-rotation circuit.