Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2504.09133

Compact Circuits for Constrained Quantum Evolutions of Sparse Operators

Franz Georg Fuchs, Ruben Pariente Bassa

Published Apr 12, 2025• 1 citation
arXivHTMLPDF

Abstract

We introduce a general framework for constructing compact quantum circuits that implement the real-time evolution of Hamiltonians of the form $H = σP_B$, where $σ$ is a Pauli string commuting with a projection operator $P_B$ 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 $\mathcal{O}(n|B|)$ CX gates and $\mathcal{O}(n |B| + \log(|B|) \log (1/ε))$ T-gates, where $n$ is the number of qubits, $|B|$ the dimension of the projected subspace, and $ε$ the desired approximation precision. For subspaces that are generated by Pauli X-orbits we further reduce complexity to $\mathcal{O}(n \log |B|)$ CX gates and $\mathcal{O}(n+\log(\frac{1}ε))$ 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 $|B|$ is small, the proposed algorithms scale favourably when compared to direct Pauli evolution.

DOI
10.48550/arxiv.2504.09133

Related papers

Scalable quantum circuits for exponential of Pauli strings and Hamiltonian simulations

Rohit Sarma Sarkar, Sabyasachi Chakraborty, Bibhas Adhikari

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 Pauli-string 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 simulation closely resembles the noiseless one when gate and idle errors are on the order of $O(10^{-3})$ or smaller.

arXiv:2405.13605•May 22, 2024•0 citations

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

Rohit Sarma Sarkar, Sabyasachi Chakraborty, Bibhas Adhikari

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.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Aug 30, 2025•1 citation

Efficient Quantum Circuits for Electronic Hamiltonian Simulation without Pauli Expansion

Tamiya Onodera, Takeshi Sato

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 $n$ spin orbitals, further expanding the optimization scope to a sequence of $O(n)$ suitably ordered triads enables a cascade of CX-gate reductions across triad boundaries, reducing the CX-gate count from $O(n^2)$ to $O(n)$. 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.

quant-pharXiv:2609.06285•Sep 8, 2026•0 citations

Tridiagonal matrix decomposition for Hamiltonian simulation on a quantum computer

Boris Arseniev, Dmitry Guskov, Richik Sengupta +2 more

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.

arXiv:2310.00121•Physical Review A•May 29, 2024•4 citations

Efficient Quantum Circuit Construction of Controlled Time-Evolution for Arbitrary Pauli-Sum Hamiltonians

Shintaro Fujiwara, Naoki Yamamoto, Naoki Ishikawa

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.

quant-pharXiv:2606.06070•Jun 5, 2026•0 citations

Sparse-QITE: Discovering Compact Generators for Quantum Imaginary-Time Evolution via Regularization Methods

Rafael Gómez-Lurbe, Luca Ion, Alberto Acevedo +1 more

Quantum imaginary-time evolution (QITE) provides a systematic route to ground-state preparation, but its cost grows rapidly with the locality $D$ of the unitary approximation, for which the local Pauli basis scales as $4^D$. We introduce Sparse-QITE, a regularized formulation that promotes compact generators, reducing the number of quantum operations and yielding circuits with fewer gates and lower depth. We derive rigorous bounds on the sparsification error and benchmark the method in one-dimensional transverse-field Ising and antiferromagnetic Heisenberg models. Sparse-QITE exhibits a controllable sparsity--accuracy trade-off while retaining strong compression even in the high-accuracy regime. In scalability experiments with respect to $D$, the active support grows much more slowly than in standard QITE and, in several regimes, approximately saturates over the domain sizes studied. After circuit construction and transpilation, some of the largest-domain cases require only about $1\%$ of the standard-QITE gate count while maintaining high accuracy. We further introduce adaptive and active-discovery variants: the former selects the regularization strength dynamically during the evolution, while the latter reduces the number of Pauli expectation values that need to be evaluated and, consequently, the measurement requirements. These results establish sparsity-promoting regularization as a route toward more resource-efficient QITE. More broadly, these results suggest that sparsity-promoting regularization may also benefit variational quantum algorithms.

quant-pharXiv:2610.20791•Oct 16, 2026•0 citations
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.
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 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.
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.
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)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.
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.
Quantum imaginary-time evolution (QITE) provides a systematic route to ground-state preparation, but its cost grows rapidly with the locality DDD of the unitary approximation, for which the local Pauli basis scales as 4D4^D4D. We introduce Sparse-QITE, a regularized formulation that promotes compact generators, reducing the number of quantum operations and yielding circuits with fewer gates and lower depth. We derive rigorous bounds on the sparsification error and benchmark the method in one-dimensional transverse-field Ising and antiferromagnetic Heisenberg models. Sparse-QITE exhibits a controllable sparsity--accuracy trade-off while retaining strong compression even in the high-accuracy regime. In scalability experiments with respect to DDD, the active support grows much more slowly than in standard QITE and, in several regimes, approximately saturates over the domain sizes studied. After circuit construction and transpilation, some of the largest-domain cases require only about 1%1\%1% of the standard-QITE gate count while maintaining high accuracy. We further introduce adaptive and active-discovery variants: the former selects the regularization strength dynamically during the evolution, while the latter reduces the number of Pauli expectation values that need to be evaluated and, consequently, the measurement requirements. These results establish sparsity-promoting regularization as a route toward more resource-efficient QITE. More broadly, these results suggest that sparsity-promoting regularization may also benefit variational quantum algorithms.