High-precision and low-depth quantum algorithm design for eigenstate problems

TL;DR

This work introduces a full-stack quantum algorithm for estimating eigenenergies and eigenstate properties that blends spectral-filter theory with randomised composite LCU techniques. By decomposing nonunitary spectral filters into hierarchical LCUs and compensating Trotter errors, the method achieves a gate complexity scaling of $ ilde{O}(\log \varepsilon^{-1})$ per circuit and near-optimal depth for lattice Hamiltonians under nearest-neighbour connectivity, while maintaining low circuit-assembly overhead. The approach supports both observable-property estimation and eigenenergy recovery, with ancilla-free measurement options that exploit conserved symmetries to reduce depth, and provides detailed resource estimates for lattice and molecular Hamiltonians, including practical IBM-device experiments achieving high-precision results. Compared to QPE, QSP, and related coherent methods, this randomised LCU framework offers strong precision scaling with favorable circuit-depth behavior under hardware constraints, making it a compelling path toward early quantum usefulness in NISQ-to-FTQC regimes.

Abstract

Estimating the eigenstate properties of quantum systems is a long-standing, challenging problem for both classical and quantum computing. Existing universal quantum algorithms typically rely on ideal and efficient query models (e.g. time evolution operator or block encoding of the Hamiltonian), which, however, become suboptimal for actual implementation at the quantum circuit level. Here, we present a full-stack design of quantum algorithms for estimating the eigenenergy and eigenstate properties, which can achieve high precision and good scaling with system size. The gate complexity per circuit for estimating generic Hamiltonians' eigenstate properties is $\tilde{O} (\log \varepsilon^{-1})$, which has a logarithmic dependence on the inverse precision $\varepsilon$. For lattice Hamiltonians, the circuit depth of our design achieves near-optimal system-size scaling, even with local qubit connectivity. Our full-stack algorithm has low overhead in circuit compilation, which thus results in a small actual gate count (CNOT and non-Clifford gates) for lattice and molecular problems compared to advanced eigenstate algorithms. The algorithm is implemented on IBM quantum devices using up to 2,000 two-qubit gates and 20,000 single-qubit gates, and achieves high-precision eigenenergy estimation for Heisenberg-type Hamiltonians, demonstrating its noise robustness.

Figures (11)

• Figure 1: Workflow of the eigenstate property and eigenenergy estimation algorithm, from formula construction to circuit implementation. (A) The hierarchical structure and components of randomised composite LCU formulae. (a1) The spectral filter is decomposed into a combination of unitaries $U(t_i) = e^{-iHt_i}$, as described by \ref{['eq:randomLCU_main']}. (a2) The real-time evolution is divided into $\nu$ segments. Each segment with time duration $t_i/\nu$ is realised by the Trotter formula $S$ and the Trotter remainder $V$ which is further decomposed into an LCU form in \ref{['eq:TrotterError_LCU_V']}, resulting in an overall composite LCU form. (a3) The overall explicit composite LCU form of $g_{\tau}$. The summands of the circuit are shown here; each circuit instance is sampled and described by \ref{['eq:U_ti_LCU']} with sampled $\vec{{i}}$. Generally, $W_{i_q}$ is composed of a Pauli rotation $R = \exp(-i \theta_{i_q} P_{i_q})$ and a tensor product of single-qubit Pauli operators, where $P_{i_q}$ is drawn from the specified probability distribution. The realisation of controlled-$W$ operation is shown in (c3). In the ancilla-free measurement scheme illustrated in (c2), $W_{i_q}$ is chosen to be a symmetry-conserved operator. (B) Illustration of spectral filtering for effective eigenstate preparation and eigenenergy search. (b1) The spectral weight of the effective state after applying the spectral filter $g_{\tau}(H-E_j)$. The $y$ axis represents the spectral weight $\braket{u_i| g_{\tau}(H-E_0)|\psi_0}$, where $|u_i\rangle$ is the eigenstate with energy $E_i$. By increasing the imaginary time $\tau$, only the spectral weight on the ground state will be preserved while the contributions from the other eigenstates are suppressed exponentially with $\tau$. (b2) The eigenenergy search using $D_{\tau}(\omega) = \braket{\psi_0 | g_{\tau}^2 (H-\omega)|\psi_0}$ where $D_{\tau}(\omega)$ is defined in \ref{['eq:OND_main']}. By increasing the imaginary time $\tau$, the peaks become sharper (from the blue line to the red line) and the true eigenenergies can be resolved by finding the peaks. The key quantity in (b1) and (b2) is $\braket{U^{\dagger} (t_j) O U(t_i)}$ in \ref{['eq:observable_dynamics_main']}. The sampled instance in $\braket{U^{\dagger} (t_j) O U(t_i)}$ can be measured using the circuit in (C). (C) Quantum circuit implementation. (c1) Circuit for general Hamiltonians. Two extreme cases: Case I ($t_j = 0$) reduces to the one in the green box; Case II ($t_i = t_j$) reduces to the one in the orange box. Cases with $t_j < t_i$ lie between and can be implemented with (c1). (c2) Ancilla-free measurement scheme for symmetry-conserved cases. $U_p$ is the unitary operator for state preparation $\ket{\psi_0} = U_p \ket{0^{\otimes n}}$ or $\frac{1}{\sqrt{2}}(\ket{\psi_{\textrm{Ref}}} + \ket{\psi_0}) = U_p \ket{0^{\otimes n}}$ as discussed in Methods. (c3) Circuit compilation for controlled $S$ and $W$, which can both be efficiently compiled into single-qubit Pauli rotation gates (green box), cnot gates and Pauli gates (blue box).
• Figure 2: The implementation of the controlled exponentiation of potential terms Ctrl-$e^{-i x \hat{V}}$ on a nearest-neighbour architecture (1D or 2D) using $[ \frac{n}{2} ]$ ancillas with depth $\mathcal{O}(n)$. (A) Copy the classical information on the qubit $A_1$ to $A_2$,...$A_{[n/2]}$, and then undo the copy operation. This circuit is equivalent to using $A_1$ as the single-controlled qubit to control all the other physical qubits $Q_1$ to $Q_n$. $Q_i$: $i$th physical qubit (encoding the $n$th spin-orbital). $A_i$: $i$th ancilla. The copy operation allows all controlled rotations (blue-shaded box in (B)) to be implemented using nearest-neighbour gates. (B) The circuit block for controlled $e^{-i x \hat{V}}$, using nearest-neighbour operations $e^{- i Z_i Z_{i+1}}$ followed by $[n/2]$swap operations detailed in C. The circuit block in (B) is repeated $\mathcal{O}(n)$ times. An example is shown in (C). cnot operations are omitted in the sub-figure. (C) 1D linear architecture. The ancillas $A_i$ ($i = 1,2,.., [n/2]$) and physical qubits $Q_i$ ($i = 1,2,.., n$) can be placed in the way shown in Step 1 in $\mathcal{O}(n)$ depth. The red arrow connecting $A_i$ and $Q_{j}$ is used to represent to perform the controlled-$\textrm{R}_z$ rotation, sandwiched by cnot operations on adjacent qubits $Q_j$ and $Q_k$ (connected and illustrated by the black arrow), which realises $e^{- i Z_j Z_{k}}$. The black arrow connecting $Q_j$ and $Q_k$ is used to represent performing the corresponding cnot operations in realising $e^{- i Z_j Z_{k}}$, then followed by a swap operation. The transformation from Step 1 (the shaded blue and orange boxes) to Step 2 can be realised by 2 swap gates (cyclic swap operation). The rest of the transformation is realised in the same way. (D) 2D planar architecture. The qubit connectivity is represented by the orange dashed line.
• Figure 3: Gate count estimates for the eigenstate property estimation tasks for the Heisenberg Hamiltonian and P450 molecule. (A) Gate count comparison with different target precision for $20$-site Heisenberg Hamiltonian. (B) Gate count comparison with increasing system size to achieve precision $0.001$. The energy gaps are determined through numerical fitting, which agrees well with the results obtained from exact diagonalisation. (C) Gate count for the P450 with $A$-type active space as a function of the energy gap, which is treated as an independent variable to analyse its effect. The pairing orders with both $k = 0$ and $1$ are shown in (C).
• Figure 4: The scaling of the T gate count with system size for the Heisenberg model. The RLCU method involves at most $10^4$ single-qubit Pauli rotation $R_z$ gates and $10^6$ T gates for $20$-qubit Heisenberg model. The circuit synthesis method is detailed in Supplementary Sec. V.
• Figure 5: Implementation of the RLCU algorithm on IBM quantum devices. We consider searching the ground state energy of a 12-qubit normalised anisotropic Heisenberg Hamiltonian \ref{['eq:Hamil_XXZ']} without any external field. We present the ideal result with finite $\tau$ and finite cutoff but without any Trotterisation error, represented by the orange dotted lines. We also show the results obtained using the noiseless and noisy Trotterised quantum circuit and the experimental data, denoted by the blue and red lines, respectively. For different lines, $D_{\tau} (\omega)$ is computed classically using data points obtained from different setups, including both numerical simulations and experimental measurements. The right panel is a zoom-in of a smaller range of ground-state energy estimates shown in the left panel. The red dotted line represents the experimentally estimated ground-state energy, which is extremely close to the ideal value shown by the black dotted line, with an error of 0.001. The energy estimation error for the excited state is about 0.005.

Theorems & Definitions (33)

• Proposition 1: Composite LCU
• Theorem 1: Gate complexity for general cases (Informal)
• Theorem 2: Gate and depth complexity for lattice Hamiltonians (Informal)
• Proposition 2: Observable estimation using the composite LCU formula
• Proposition 3: Elementary gate count
• Proposition 4
• Theorem 3: Eigenstate property estimation for generic Hamiltonians (formal version of \ref{['thm:observ_estimation_main']})
• Theorem 4: Gate and depth complexity for lattice Hamiltonians
• proof
• Proposition 5: Composite LCU formula in a continuous form