Quantum-compatible Riemannian conjugate gradient for ground-state preparation

The gist The proposed approach, Quantum Conjugate Gradient Descent (QCGD), extends Riemannian optimization to quantum ground-state preparation by combining steepest descent with information from previous search directions, achieving faster convergence per iteration than baseline methods like DB-QITE

Geometric Formulation for Ground-State Preparation

Riemannian optimization provides a natural geometric framework for designing quantum algorithms for ground-state preparation, most notably through quantum imaginary-time evolution The problem of lowering the energy of an initial state, 0⟩, with respect to a Hamiltonian H can be formulated as the minimisation of E: U(N) → R on the unitary group U(N) This minimization involves taking each step along geodesics connecting two points on the manifold Uk and Uk+1 The Riemannian gradient of a function f: M → R at x ∈ M is the unique tangent vector grad f(x) ∈ TxM satisfying ⟨grad f(x), η⟩x = Df(x)[η] for every tangent vector η ∈ TxM

Double-Bracket Quantum Imaginary-Time Evolution

DB-QITE tackles the open problem of designing efficient quantum circuits guided by imaginary-time evolution (ITE) It exploits the relation between ITE and double bracket flows to propose a non-variational, systematic approach From a geometric perspective, DBQITE is understood as a quantum-compatible implementation of the GD iteration defined in Eq. (9) The update rule for DB-QITE is given by Uk+1 = e−µk[H,ψk]Uk, where µk > 0 is selected by a line search to minimise E(Uk+1) and ψk = Ukρ0U†k

Overview of Riemannian Conjugate Gradient Descent

Classical CGD methods are among the most successful first-order techniques in classical optimization and can be extended to any Riemannian manifold M The generalized iteration sequence is given by xk+1 = Rxk(tkηk) For the specific problem, the Riemannian CGD scheme is defined as Uk+1 = (e−µk(−[H,ψk]+γkWk−1)Uk k > 0, e−µ0[H,ψ0]U0 k = 0 This update allows for a larger per-step energy decrease than GD both in Euclidean space and on Riemannian manifolds such as U(N)

Proposed Method: Quantum Conjugate Gradient Descent

QCGD generates a sequence of unitaries U QCGD k and associated states ψ QCGD k⟩:= U QCGD k0⟩ The search direction is defined by W k = −[H, ϕ k], which expands to Wk = −[H, ψk] − γk[H, ψk−1] − γkγk−1[H, ψk−2] - · · · − γk · · · γ1[H, ψ0] The resulting update formula is Uk+1 = e i√µ kHe i√µ kckψ k·.. e iθ0ψ0 e −i√µ kHe −iθ0ψ0·.. e −iθk−1ψk-1Uk, where U QCGD k is produced

Numerical Results and Comparison

Numerical simulations show that QCGD consistently improves energy reduction and ground-state fidelity across the tested trajectories For L = 20 with HVA initialisation, QCGD produces a larger energy reduction at the second update and retains this accumulated advantage In terms of compiled circuit resources, for L = 12, two QCGD iterations match or exceed the fidelity obtained after three DB-QITE updates This is demonstrated by Table III, where two QCGD updates achieved F2 = 0.955, slightly exceeding the final DB-QITE value F3 = 0.953

Conclusion

The results establish that memory-based Riemannian optimization can translate into a substantial state-preparation circuit advantage for quantum GSP QCGD is shown to produce both faster convergence per iteration and a substantial resource advantage at matched fidelity Natural next steps include testing the robustness of the matched-fidelity resource advantage in broader settings, such as different simulation settings and potential algorithmic modifications Overall, the present results establish QCGD as a concrete demonstration that memory-based Riemannian optimization can translate into a substantial state-preparation circuit advantage for quantum GSP >

--- Page 10 ---

The gist The proposed approach, Quantum Conjugate Gradient Descent (QCGD), extends Riemannian optimization to quantum ground-state preparation by combining steepest descent with information from previous search directions, achieving faster convergence per iteration than baseline methods like DB-QITE

Numerical Results and Comparison

Numerical simulations show that QCGD consistently improves energy reduction and ground-state fidelity across the tested trajectories For L = 20 with HVA initialisation, QCGD produces a larger energy reduction at the second update and retains this accumulated advantage In terms of compiled circuit resources, for L = 12, two QCGD iterations match or exceed the fidelity obtained after three DB-QITE updates This is demonstrated by Table III, where two QCGD updates achieved F2 = 0.955, slightly exceeding the final DB-QITE value F3 = 0.

1. Improve ground-state preparation (GSP) for NP-hard/QMA-complete problems by implementing a memory-augmented search direction that can achieve a larger per-step energy decrease than GD both in Euclidean space and on Riemannian manifolds such as U(N) by leveraging the conjugate gradient property. This allows the AI system to converge toward the ground state faster, especially in early fault-tolerant regimes where circuit depth growth is manageable.

2. Implement a quantum algorithm that can match or improve the fidelity with one fewer step compared to double-bracket quantum imaginary-time evolution (DB-QITE) for specific Hamiltonians and initializations. This means the AI system can reach a target energy/fidelity threshold using fewer total circuit updates, providing a practical advantage in terms of overall computational resource usage at matched fidelity.

3. Develop an AI system that optimizes state preparation by dynamically selecting conjugate parameters (µk, γk) based on the energy reduction landscape, even when restricted to finite search domains. This allows the AI to navigate complex optimization problems more effectively than standard gradient descent by exploiting the memory effect inspired by conjugate gradient descent (CGD).

Riemannian optimization provides a natural geometric framework for designing quantum algorithms for ground-state preparation, most notably through quantum imaginary-time evolution. However, existing geometry-aware approaches primarily employ steepest-descent updates, leaving more advanced methods from classical Riemannian optimization largely unexplored. In this work, we introduce quantum conjugate gradient descent (QCGD), a quantum-compatible implementation of Riemannian conjugate gradient that combines the steepest descent direction with information retained from previous search directions. In our numerical experiments, the proposed approach achieves faster convergence per iteration than the baseline double-bracket quantum imaginary-time evolution (DB-QITE) algorithm, and we quantify the associated circuit cost with gate counts produced by Qrisp. This improvement, however, comes at the cost of increased circuit depth for each update step, revealing a trade-off between convergence rate and implementation complexity. Nevertheless, since the circuit depth for those algorithms grows exponentially with the number of iterations, the proposed method may offer practical advantages, particularly in the early fault-tolerant regime.