SqDRIFT: Randomized Quantum Ground-State Estimation

• SqDRIFT is a randomized quantum algorithm that approximates ground-state energies by combining qDRIFT’s stochastic simulation with SKQD subspace construction.
• It employs shallow circuits on near-term devices to efficiently simulate many-body quantum systems while maintaining provable error bounds.
• The method leverages classical subspace diagonalization to scale quantum chemistry applications without increasing circuit depth.

SqDRIFT is a randomized quantum algorithm designed to efficiently approximate ground-state energies of many-body quantum systems, particularly electronic structure Hamiltonians, on near-term and early fault-tolerant quantum devices. The method combines Sample-based Krylov Quantum Diagonalization (SKQD) with qDRIFT, a stochastic Hamiltonian simulation approach, yielding an algorithm with provable convergence guarantees and circuit-resource requirements compatible with hardware constraints on modern superconducting processors (Piccinelli et al., 4 Aug 2025, Piccinelli et al., 9 Mar 2026).

1. Algorithmic Foundation

SqDRIFT targets the ground-state energy estimation for an $n$-qubit Hamiltonian,

$$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$

where the ground state $|\phi_0\rangle$ is assumed to be "concentrated"—i.e., supported on $L \ll 2^n$ computational basis states. The method employs the SKQD framework, which constructs a Krylov subspace,

$$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$

based on repeated, imperfect time evolution from an initial reference, typically a Hartree–Fock state.

SqDRIFT replaces exact or Trotterized time evolution (requiring excessive circuit depth for large, complex systems) with stochastic, shallow qDRIFT circuits. Each Krylov vector is generated via multiple independent qDRIFT realizations, and computational-basis measurements are drawn from each. The union of all unique observed bitstrings is used to construct a much smaller subspace (typically $L\sim \mathrm{poly}(n)$), into which the Hamiltonian is projected and diagonalized classically to yield the approximate ground-state energy (Piccinelli et al., 4 Aug 2025).

2. Mathematical Structure and Convergence Guarantees

2.1 qDRIFT Approximation

For

$H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$

one defines $\lambda = \sum_i |c_i|$ and constructs a stochastic sequence of $N$ exponentials:

$$V_{\mathbf{k}} = \prod_{j=1}^N \exp\left(-i h_{k_j} \frac{\lambda t}{N}\right),\quad \Pr[k_j = i] = \frac{|c_i|}{\lambda}.$$

The average quantum channel satisfies

$$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$0

independent of the number of Hamiltonian terms, allowing control of the simulation error by tuning $$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$1. For simulation error $$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$2,

$$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$3

2.2 Krylov Subspace and Sampling

A $$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$4-step Krylov subspace is realized through repeated application of randomized qDRIFT unitaries, each followed by $$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$5 measurements. The ensemble of bitstrings from multiple ($$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$6) independent circuits provides a set of computational-basis Slater determinants spanning the subspace.

2.3 Convergence Theorem and Sampling Bounds

Given ground-state concentration parameters $$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$7:

$$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$8

with $$H = \sum_{i=1}^{\mathcal N} c_i h_i, \quad \lambda = \sum_i |c_i|,$$9, the lowest-energy vector $|\phi_0\rangle$0 in the subspace has variational error

$|\phi_0\rangle$1

The probability of failing to sample all $|\phi_0\rangle$2 important bitstrings,

$|\phi_0\rangle$3

can be controlled by increasing $|\phi_0\rangle$4, $|\phi_0\rangle$5, and appropriate choice of $|\phi_0\rangle$6 (Piccinelli et al., 4 Aug 2025).

3. Practical Workflow and Implementation

3.1 Pseudocode Overview

The SqDRIFT procedure can be summarized as follows:

$N$5 (Piccinelli et al., 4 Aug 2025, Piccinelli et al., 9 Mar 2026)

3.2 Hamiltonian Mapping

Electronic structure Hamiltonians are mapped to $|\phi_0\rangle$7-qubit Pauli string sums via Jordan–Wigner or Bravyi–Kitaev transformations,

$|\phi_0\rangle$8

with $|\phi_0\rangle$9. The procedure is fully general for arbitrary sparse sums.

4. Resource Requirements and Complexity

Resource	Scaling	Quantities Used (Typical)
Circuit depth	$L \ll 2^n$0, with $L \ll 2^n$1	$L \ll 2^n$2–$L \ll 2^n$3 for PAHs; $L \ll 2^n$4 for 100 qubits
Krylov depth	$L \ll 2^n$5	$L \ll 2^n$6 (PAHs); $L \ll 2^n$7 (half-Möbius study)
Number of circuits	$L \ll 2^n$8	$L \ll 2^n$9–$$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$0
Measurements	$$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$1	$$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$2–$$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$3

• Total two-qubit gate depth per circuit is $$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$4, e.g., $$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$5 for $$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$6-qubit experiments.
• Classical overhead for the subspace diagonalization scales as $$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$7 where $$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$8 is the number of unique sampled bitstrings (typically $$\mathcal{K}_d = \text{span}\left\{|\psi_k\rangle = \left[e^{-i H t}\right]^k |\psi_0\rangle \right\}_{k=0}^{d-1},$$9).
• The per-circuit error is set by $L\sim \mathrm{poly}(n)$0, independent of Hamiltonian term count.
• Active-space size can be increased without additional quantum resources, as $L\sim \mathrm{poly}(n)$1 and $L\sim \mathrm{poly}(n)$2 are fixed for fixed target error (Piccinelli et al., 4 Aug 2025, Piccinelli et al., 9 Mar 2026).

5. Representative Applications and Empirical Results

5.1 Polycyclic Aromatic Hydrocarbons (PAHs)

• Naphthalene ($L\sim \mathrm{poly}(n)$3 qubits, $L\sim \mathrm{poly}(n)$4, $L\sim \mathrm{poly}(n)$5, $L\sim \mathrm{poly}(n)$6, $L\sim \mathrm{poly}(n)$7 or $L\sim \mathrm{poly}(n)$8): Achieved sub-millihartree energy error, surpassing CISD, using subspaces of $L\sim \mathrm{poly}(n)$9 determinants per run.
• Coronene ($H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$0 qubits, $H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$1, $H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$2, $H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$3, $H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$4–$H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$5): Subspace dimensions $H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$6–$H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$7, variational energy within $H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$8 Ha of FCIQMC, outperforming CISD and approaching HCI accuracy (Piccinelli et al., 4 Aug 2025).

5.2 Large-Scale Molecules on Superconducting Hardware

• Half-Möbius molecule: Simulations on $H = \sum_i c_i h_i, \quad \|h_i\| \leq 1,$9–$\lambda = \sum_i |c_i|$0 qubits, fixed subspace size $\lambda = \sum_i |c_i|$1, $\lambda = \sum_i |c_i|$2, $\lambda = \sum_i |c_i|$3.
• Gate fidelities of $\lambda = \sum_i |c_i|$4–$\lambda = \sum_i |c_i|$5\%, readout fidelity $\lambda = \sum_i |c_i|$6\%. Error mitigation achieved further improvement.
• Correlation energy for $\lambda = \sum_i |c_i|$7 orbitals ($\lambda = \sum_i |c_i|$8 qubits) recovered $\lambda = \sum_i |c_i|$9 Ha relative to Hartree–Fock, with resource requirements stable as active space grows (Piccinelli et al., 9 Mar 2026).

6. Scaling Properties and Hardware Compatibility

SqDRIFT enables utility-scale quantum chemistry by trading circuit depth for randomization and sampling—resource modalities well-matched to NISQ and early fault-tolerant hardware. The qDRIFT-generated circuits are shallow and hardware-friendly. The scaling of quantum resources is polylogarithmic in system size for fixed error threshold and fixed ground-state concentration, and essentially independent of Hilbert space dimension.

Classical post-processing (determinant selection, subspace Hamiltonian diagonalization) remains subdominant and is readily managed on modest HPC resources even at $N$0 (Piccinelli et al., 4 Aug 2025, Piccinelli et al., 9 Mar 2026).

7. Significance and Outlook

SqDRIFT demonstrates that quantum-assisted electronic structure calculations can be performed with circuit depths compatible with near-term device capabilities while maintaining provable convergence. Scaling studies confirm that larger active spaces yield systematically improved correlation energies without increased circuit depth, establishing a viable pathway toward quantum simulations of previously intractable chemical systems. With further progress in gate fidelity and coherence, SqDRIFT is projected to be applicable to active spaces of $N$1–$N$2 orbitals ($N$3–$N$4 qubits) and beyond, extending the reach of quantum computational chemistry well past the limits of classical exact diagonalization (Piccinelli et al., 4 Aug 2025, Piccinelli et al., 9 Mar 2026).