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:2604.02569

RFOX (Rotated-Field Oscillatory eXchange) quantum algorithm: Towards Parameter-Free Quantum Optimizers

Brian García Sarmina, Guo-Hua Sun, Shi-Hai Dong

On Aqora:SarminaSarmina
quant-phmath.OCSubmitted Apr 2, 2026• Announced Apr 6, 2026• 0 citations• Revised May 19, 2026 (v3)
v1 · Apr 2, 2026v2 · May 6, 2026v3 · May 19, 2026
arXivHTMLPDF

Abstract

We introduce RFOX (Rotated-Field Oscillatory eXchange), a parameter-free quantum algorithm for combinatorial optimization that combines an almost constant non-stoquastic $XX$ catalyst with a weak harmonic $ZX$ counter-diabatic term. Using the Floquet-Magnus expansion, we derive an effective Hamiltonian whose leading-order $\mathcal{O}(δ/ω)$ corrections yield local $Y$ fields, field-modulated 2-body terms, and poly-local 3-body topological interactions driven by graph connectivity. This structure ensures a nearly flat instantaneous spectral gap, preventing the unpredictable gap collapses typical of conventional $X$ (stoquastic), $XX$, and $X+sXX$ (non-stoquastic) driver schedules. Extensive noiseless simulations and physical hardware experiments on IBM Quantum processors (up to 20 qubits) validate our spectral predictions. RFOX consistently attains near-optimal or exact ground states in the random-field Ising model using up to an order of magnitude fewer Trotter slices, with an advantage that grows alongside problem disorder. These results suggest that fixed-gap, non-stoquastic drivers augmented with analytically derived counter-diabatic terms offer a scalable, tuning-free route for quantum optimization.

Comments
21 pages, 14 figures

Related papers

Essentiality of the Non-stoquastic Hamiltonians and Driver Graph Design in Quantum Optimization Annealing

Vicky Choi

One of the distinct features of quantum mechanics is that the probability amplitude can have both positive and negative signs, which has no classical counterpart as the classical probability must be positive. Consequently, one possible way to achieve quantum speedup is to explicitly harness this feature. Unlike a stoquastic Hamiltonian whose ground state has only positive amplitudes (with respect to the computational basis), a non-stoquastic Hamiltonian can be eventually stoquastic or properly non-stoquastic when its ground state has both positive and negative amplitudes. In this paper, we describe that, for some hard instances which are characterized by the presence of an anti-crossing (AC) in a transverse-field quantum annealing (QA) algorithm, how to design an appropriate XX-driver graph (without knowing the prior problem structure) with an appropriate XX-coupler strength such that the resulting non-stoquastic QA algorithm is proper-non-stoquastic with two bridged anti-crossings (a double-AC) where the spectral gap between the first and second level is large enough such that the system can be operated diabatically in polynomial time. The speedup is exponential in the original AC-distance, which can be sub-exponential or exponential in the system size, over the stoquastic QA algorithm, and possibly the same order of speedup over the state-of-the-art classical algorithms in optimization. This work is developed based on the novel characterizations of a modified and generalized parametrization definition of an anti-crossing in the context of quantum optimization annealing introduced in [4].

arXiv:2105.02110•May 5, 2021•16 citations

Bias-field digitized counterdiabatic quantum algorithm for higher-order binary optimization

Sebastián V. Romero, Anne-Maria Visuri, Alejandro Gomez Cadavid +3 more

Abstract Combinatorial optimization plays a crucial role in many industrial applications. While classical computing often struggles with complex instances, quantum optimization emerges as a promising alternative. Here, we present an enhanced bias-field digitized counterdiabatic quantum optimization (BF-DCQO) algorithm to address higher-order unconstrained binary optimization (HUBO). We apply BF-DCQO to a HUBO problem featuring three-local terms in the Ising spin-glass model, validated experimentally using 156 qubits on an IBM quantum processor. In the studied instances, our results outperform standard methods such as the quantum approximate optimization algorithm, quantum annealing, simulated annealing, and Tabu search. Furthermore, we provide numerical evidence of the feasibility of a similar HUBO problem on a 433-qubit Osprey-like quantum processor. Finally, we solve denser instances of the MAX 3-SAT problem in an IonQ emulator. Our results show that BF-DCQO offers an effective path for solving large-scale HUBO problems on current and near-term quantum processors.

arXiv:2409.04477•Communications Physics•Aug 23, 2025•32 citations

Greedy parameter optimization for diabatic quantum annealing

Tadashi Kadowaki, Hidetoshi Nishimori

A shorter processing time is desirable for quantum computation to minimize the effects of noise. We propose a simple procedure to variationally determine a set of parameters in the transverse-field Ising model for quantum annealing (QA) appended with a field along the [Formula: see text]-axis. The method consists of greedy optimization of the signs of coefficients of the [Formula: see text]-field term based on the outputs of short annealing processes. We test the idea in the ferromagnetic system with all-to-all couplings and spin-glass problems, and find that the method outperforms the traditional form of QA and simulated annealing in terms of the success probability and the time to solution, in particular, in the case of shorter annealing times, achieving the goal of improved performance while avoiding noise. The non-stoquastic [Formula: see text] term can be eliminated by a rotation in the spin space, resulting in a non-trivial diabatic control of the coefficients in the stoquastic transverse-field Ising model, which may be feasible for experimental realization. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

arXiv:2111.13287•Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Dec 4, 2022•15 citations

Basis Adaptive Algorithm for Quantum Many-Body Systems on Quantum Computers

Anutosh Biswas, Sayan Ghosh, Ritajit Majumdar +3 more

We introduce a Basis Adaptive (BA) algorithm for hybrid quantum-classical simulation of correlated quantum many-body systems. Starting from a small set of physically motivated bitstrings, the algorithm iteratively applies a single-step first-order Trotterized circuit on a quantum processor, filters the sampled configurations by enforcing $U(1)$ spin conservation and lattice reflection symmetry, and classically diagonalizes the Hamiltonian in the resulting reduced Hilbert space. This design avoids the variational optimization overhead of VQE, the deep coherent circuits required by QPE, and the symmetry-violating subspaces that arise in SKQD. The ground-state energy error is bounded analytically by $\sqrt{8}\,\|H\|\left(1-\sqrt{α_{D_T}}\right)^{1/2}$, where $α_{D_T}$ is the probability weight captured by the $D_T$ sampled basis states. This bound connects algorithm performance directly to ground-state sparsity and explains the observed accuracy hierarchy across different phases. Benchmarked on the spin-$1/2$ Heisenberg XXZ chain (up to $N=62$ qubits on the IBM Heron processor), the algorithm achieves a $3.5\%$ energy error in the gapped Neel phase ($Δ=2.0$) and below $0.5\%$ at the ferromagnetic boundary ($Δ=-1.0$). The accuracy degrades to $28.7\%$ in the strongly quasi-long-range-ordered regime ($Δ=0.5$). Spin-spin correlation functions are reproduced across all regimes, confirming that symmetry-filtered real-time sampling provides a practical and noise-resilient pathway to ground-state properties on near-term quantum hardware.

cond-mat.str-elCross-listedarXiv:2512.12753•Dec 16, 2025•0 citations

Quantum resources in non-stoquastic quantum annealing

Chiara Capecci, Sebastian Nagies, Naga Dileep Varikuti +1 more

Quantum annealing promises to solve combinatorial optimization problems by preparing the ground state of a target Hamiltonian. Standard annealing protocols are, however, stoquastic and can thus be simulated by sign-problem-free quantum Monte-Carlo methods. To obtain a true quantum advantage, it has been proposed to use non-stoquastic catalyst Hamiltonians. Active only at intermediate stages of the protocol, these can, for certain problems, convert first-order into second-order quantum phase transitions and thus permit an exponential speedup over the stoquastic protocol. At the same time, the non-stoquastic catalyst renders quantum Monte-Carlo methods inefficient. It remains, however, an open question how other classical computation methods are affected by the non-stoquastic terms. We address this question by computing quantum resources -- entanglement entropy and stabilizer Rényi entropy -- whose presence makes classical computations based on tensor networks and stabilizer-tableau methods exponentially hard. We compare these with the spectral gap along the annealing path for two paradigmatic benchmark models, the fully connected $p$-spin model and a geometrically local Ising model. While the exact behavior shows a subtle dependency on the underlying model and the annealing path, our numerics suggest consistently that the scaling of entanglement and non-stabilizerness is at least maintained in the deeply non-stoquastic regime and in some cases even significantly enhanced. Our results thus suggest that improvements of quantum performance in non-stoquastic annealing coincide with significant presence of quantum computational resources.

quant-pharXiv:2606.09995•Jun 10, 2026•3 citations

Snapshot-QAOA: Extending QAOA to Quantum Hamiltonian Simulation

Reuben Tate, Quinn Langfitt, Elijah Pelofske +4 more

Abstract We present Snapshot-QAOA, a variation of the Quantum Approximate Optimization Algorithm (QAOA) that finds approximate minimum energy eigenstates of a large set of quantum Hamiltonians. Traditionally, QAOA targets the task of approximately solving combinatorial optimization problems. Snapshot-QAOA enables a significant expansion of the use case space for QAOA to more general quantum Hamiltonians, where the goal is to approximate the ground-state. Snapshot-QAOA retains the desirable variational-algorithm qualities of QAOA such as a small parameter count and relatively shallow circuit depth. Snapshot-QAOA is thus a better trainable alternative to the (Near Intermediate-Scale Quantum) NISQ-era Variational Quantum Eigensolver (VQE) algorithm, while retaining a significant circuit-depth advantage over the (Quantum Error Corrected) QEC-era Quantum Phase Estimation (QPE) algorithm. Our fundamental approach is inspired by the idea of Trotterization of a continuous-time linear adiabatic anneal schedule. Snapshot-QAOA restricts the QAOA evolution to not phasing out the mixing Hamiltonian completely at the end of the evolution, instead evolving only a partial typical linear QAOA schedule, thus creating a type of snapshot of the typical QAOA evolution. In this way, the cost of the quantum Hamiltonian is encoded by the phase separator to address the diagonal terms in the problem Hamiltonian and by the mixer to address the non-diagonal terms in the problem Hamiltonian. We focus on QAOA with the transverse field mixer, and thus simulations of quantum Hamiltonians that contain single-site Pauli-X terms. By measuring the expectation value of the system in both X and Z-basis, we can estimate the ground-state energy of transverse field quantum Hamiltonians. The accuracy of ground-state energy finding with snapshot-QAOA is evaluated using extensive numerical simulations on a 16 qubit J1-J2 frustrated transverse field Ising model.

•Mar 22, 2026•0 citations
We introduce RFOX (Rotated-Field Oscillatory eXchange), a parameter-free quantum algorithm for combinatorial optimization that combines an almost constant non-stoquastic XXXXXX catalyst with a weak harmonic ZXZXZX counter-diabatic term. Using the Floquet-Magnus expansion, we derive an effective Hamiltonian whose leading-order O(δ/ω)\mathcal{O}(δ/ω)O(δ/ω) corrections yield local YYY fields, field-modulated 2-body terms, and poly-local 3-body topological interactions driven by graph connectivity. This structure ensures a nearly flat instantaneous spectral gap, preventing the unpredictable gap collapses typical of conventional XXX (stoquastic), XXXXXX, and X+sXXX+sXXX+sXX (non-stoquastic) driver schedules. Extensive noiseless simulations and physical hardware experiments on IBM Quantum processors (up to 20 qubits) validate our spectral predictions. RFOX consistently attains near-optimal or exact ground states in the random-field Ising model using up to an order of magnitude fewer Trotter slices, with an advantage that grows alongside problem disorder. These results suggest that fixed-gap, non-stoquastic drivers augmented with analytically derived counter-diabatic terms offer a scalable, tuning-free route for quantum optimization.
One of the distinct features of quantum mechanics is that the probability amplitude can have both positive and negative signs, which has no classical counterpart as the classical probability must be positive. Consequently, one possible way to achieve quantum speedup is to explicitly harness this feature. Unlike a stoquastic Hamiltonian whose ground state has only positive amplitudes (with respect to the computational basis), a non-stoquastic Hamiltonian can be eventually stoquastic or properly non-stoquastic when its ground state has both positive and negative amplitudes. In this paper, we describe that, for some hard instances which are characterized by the presence of an anti-crossing (AC) in a transverse-field quantum annealing (QA) algorithm, how to design an appropriate XX-driver graph (without knowing the prior problem structure) with an appropriate XX-coupler strength such that the resulting non-stoquastic QA algorithm is proper-non-stoquastic with two bridged anti-crossings (a double-AC) where the spectral gap between the first and second level is large enough such that the system can be operated diabatically in polynomial time. The speedup is exponential in the original AC-distance, which can be sub-exponential or exponential in the system size, over the stoquastic QA algorithm, and possibly the same order of speedup over the state-of-the-art classical algorithms in optimization. This work is developed based on the novel characterizations of a modified and generalized parametrization definition of an anti-crossing in the context of quantum optimization annealing introduced in [4].
Abstract Combinatorial optimization plays a crucial role in many industrial applications. While classical computing often struggles with complex instances, quantum optimization emerges as a promising alternative. Here, we present an enhanced bias-field digitized counterdiabatic quantum optimization (BF-DCQO) algorithm to address higher-order unconstrained binary optimization (HUBO). We apply BF-DCQO to a HUBO problem featuring three-local terms in the Ising spin-glass model, validated experimentally using 156 qubits on an IBM quantum processor. In the studied instances, our results outperform standard methods such as the quantum approximate optimization algorithm, quantum annealing, simulated annealing, and Tabu search. Furthermore, we provide numerical evidence of the feasibility of a similar HUBO problem on a 433-qubit Osprey-like quantum processor. Finally, we solve denser instances of the MAX 3-SAT problem in an IonQ emulator. Our results show that BF-DCQO offers an effective path for solving large-scale HUBO problems on current and near-term quantum processors.
A shorter processing time is desirable for quantum computation to minimize the effects of noise. We propose a simple procedure to variationally determine a set of parameters in the transverse-field Ising model for quantum annealing (QA) appended with a field along the [Formula: see text]-axis. The method consists of greedy optimization of the signs of coefficients of the [Formula: see text]-field term based on the outputs of short annealing processes. We test the idea in the ferromagnetic system with all-to-all couplings and spin-glass problems, and find that the method outperforms the traditional form of QA and simulated annealing in terms of the success probability and the time to solution, in particular, in the case of shorter annealing times, achieving the goal of improved performance while avoiding noise. The non-stoquastic [Formula: see text] term can be eliminated by a rotation in the spin space, resulting in a non-trivial diabatic control of the coefficients in the stoquastic transverse-field Ising model, which may be feasible for experimental realization. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
We introduce a Basis Adaptive (BA) algorithm for hybrid quantum-classical simulation of correlated quantum many-body systems. Starting from a small set of physically motivated bitstrings, the algorithm iteratively applies a single-step first-order Trotterized circuit on a quantum processor, filters the sampled configurations by enforcing U(1)U(1)U(1) spin conservation and lattice reflection symmetry, and classically diagonalizes the Hamiltonian in the resulting reduced Hilbert space. This design avoids the variational optimization overhead of VQE, the deep coherent circuits required by QPE, and the symmetry-violating subspaces that arise in SKQD. The ground-state energy error is bounded analytically by 8 ∥H∥(1−αDT)1/2\sqrt{8}\,\|H\|\left(1-\sqrt{α_{D_T}}\right)^{1/2}8​∥H∥(1−αDT​​​)1/2, where αDTα_{D_T}αDT​​ is the probability weight captured by the DTD_TDT​ sampled basis states. This bound connects algorithm performance directly to ground-state sparsity and explains the observed accuracy hierarchy across different phases. Benchmarked on the spin-1/21/21/2 Heisenberg XXZ chain (up to N=62N=62N=62 qubits on the IBM Heron processor), the algorithm achieves a 3.5%3.5\%3.5% energy error in the gapped Neel phase (Δ=2.0Δ=2.0Δ=2.0) and below 0.5%0.5\%0.5% at the ferromagnetic boundary (Δ=−1.0Δ=-1.0Δ=−1.0). The accuracy degrades to 28.7%28.7\%28.7% in the strongly quasi-long-range-ordered regime (Δ=0.5Δ=0.5Δ=0.5). Spin-spin correlation functions are reproduced across all regimes, confirming that symmetry-filtered real-time sampling provides a practical and noise-resilient pathway to ground-state properties on near-term quantum hardware.
Quantum annealing promises to solve combinatorial optimization problems by preparing the ground state of a target Hamiltonian. Standard annealing protocols are, however, stoquastic and can thus be simulated by sign-problem-free quantum Monte-Carlo methods. To obtain a true quantum advantage, it has been proposed to use non-stoquastic catalyst Hamiltonians. Active only at intermediate stages of the protocol, these can, for certain problems, convert first-order into second-order quantum phase transitions and thus permit an exponential speedup over the stoquastic protocol. At the same time, the non-stoquastic catalyst renders quantum Monte-Carlo methods inefficient. It remains, however, an open question how other classical computation methods are affected by the non-stoquastic terms. We address this question by computing quantum resources -- entanglement entropy and stabilizer Rényi entropy -- whose presence makes classical computations based on tensor networks and stabilizer-tableau methods exponentially hard. We compare these with the spectral gap along the annealing path for two paradigmatic benchmark models, the fully connected ppp-spin model and a geometrically local Ising model. While the exact behavior shows a subtle dependency on the underlying model and the annealing path, our numerics suggest consistently that the scaling of entanglement and non-stabilizerness is at least maintained in the deeply non-stoquastic regime and in some cases even significantly enhanced. Our results thus suggest that improvements of quantum performance in non-stoquastic annealing coincide with significant presence of quantum computational resources.
Abstract We present Snapshot-QAOA, a variation of the Quantum Approximate Optimization Algorithm (QAOA) that finds approximate minimum energy eigenstates of a large set of quantum Hamiltonians. Traditionally, QAOA targets the task of approximately solving combinatorial optimization problems. Snapshot-QAOA enables a significant expansion of the use case space for QAOA to more general quantum Hamiltonians, where the goal is to approximate the ground-state. Snapshot-QAOA retains the desirable variational-algorithm qualities of QAOA such as a small parameter count and relatively shallow circuit depth. Snapshot-QAOA is thus a better trainable alternative to the (Near Intermediate-Scale Quantum) NISQ-era Variational Quantum Eigensolver (VQE) algorithm, while retaining a significant circuit-depth advantage over the (Quantum Error Corrected) QEC-era Quantum Phase Estimation (QPE) algorithm. Our fundamental approach is inspired by the idea of Trotterization of a continuous-time linear adiabatic anneal schedule. Snapshot-QAOA restricts the QAOA evolution to not phasing out the mixing Hamiltonian completely at the end of the evolution, instead evolving only a partial typical linear QAOA schedule, thus creating a type of snapshot of the typical QAOA evolution. In this way, the cost of the quantum Hamiltonian is encoded by the phase separator to address the diagonal terms in the problem Hamiltonian and by the mixer to address the non-diagonal terms in the problem Hamiltonian. We focus on QAOA with the transverse field mixer, and thus simulations of quantum Hamiltonians that contain single-site Pauli-X terms. By measuring the expectation value of the system in both X and Z-basis, we can estimate the ground-state energy of transverse field quantum Hamiltonians. The accuracy of ground-state energy finding with snapshot-QAOA is evaluated using extensive numerical simulations on a 16 qubit J1-J2 frustrated transverse field Ising model.