Virtual Ground-State Prep Scheme

• Virtual ground-state preparation schemes are quantum algorithms that use ancilla-assisted LCU and QSP techniques to project an initial state onto the ground state of a Hamiltonian.
• They leverage Gaussian filtering and Fourier series representations to achieve effective projection while optimizing query complexity based on the spectral gap and state overlap.
• These methods are scalable and versatile, with extensions to excited state preparation and quadratic speedups in frustration-free systems via spectral-gap amplification.

A virtual ground-state preparation scheme refers to a family of quantum algorithms that facilitate the preparation of the ground state of a given Hamiltonian without directly implementing non-unitary (e.g., imaginary time) evolution. These protocols utilize ancilla-assisted, measurement-based, or signal-processing-based constructions to amplify the ground-state component of an initial probe state. The defining features are their ability to realize effective ground-state projectors through polynomial or Fourier series in the Hamiltonian, often using quantum signal processing (QSP), linear combinations of unitaries (LCU), or related block-encoding circuits. The primary performance metrics are query complexity in terms of inverse spectral gap, overlap with the ground state, circuit depth, and ancilla resource overhead. The approach is particularly powerful as it can saturate known lower bounds for ground-state preparation complexity and can sometimes be extended to prepare excited states, frustration-free Hamiltonians, or to boost variational methods.

1. Fundamental Construction: LCU and Gaussian Projectors

At the core of the virtual ground-state scheme is the observation that the operator $\exp(-\frac{1}{2} t^2 H^2)$, for large $t$, serves as a filter that projects onto the ground-state manifold of a normalized Hermitian Hamiltonian $H$. The action on an initial state $|\psi_0\rangle$ with ground state overlap $\gamma = |\langle \lambda_0|\psi_0\rangle|$ yields a final state $\epsilon$-close to $|\lambda_0\rangle$ (true ground state) when $t=\Theta[(1/\Delta)\sqrt{\log(1/(\gamma\eta))}]$, where $\Delta$ is the spectral gap and $\eta$ is the target vector error (Keen et al., 2021).

To realize $\exp(-\frac{1}{2} t^2 H^2)$ on a quantum circuit, the protocol uses:

• Hubbard–Stratonovich Transform: Expresses the non-unitary operator as a Gaussian-weighted continuous integral over unitary time-evolution operators:

$$\exp(-\tfrac{1}{2} t^2 H^2) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty dz\, e^{-z^2/2} e^{-i z t H}$$

• LCU Discretization: The integral is truncated to $|z|<z_c$ and discretized to a finite sum:

$$h(H) = \sum_{k=-N}^{N} \alpha_k\, U_k;\quad \alpha_k = (\Delta_z/\sqrt{2\pi}) e^{-z_k^2/2},\; U_k = e^{-i z_k t H}$$

where $z_k = k \Delta_z$, $\Delta_z = 2\pi/(z_c + t)$, and $N = \Theta([(1/\Delta) \log(1/(\gamma\eta))])$.

• Quantum Implementation: The LCU operation is synthesized by preparing an ancilla register in the state $\sum_k \sqrt{\alpha_k}\,|k\rangle$, performing a controlled-$U_k$ (Hamiltonian simulation) for each $k$, uncomputing the ancilla, and post-selecting on $|0\ldots0\rangle$ success. Amplitude amplification can boost the success probability $p_{succ} = \Omega(\gamma)$ so that the total cost is dominated by $O(1/\gamma)$ repetitions.

This scheme saturates the near-optimal scaling for generic gapped Hamiltonians: $$\text{Query complexity} = O\left(\frac{1}{\gamma\Delta} \log\frac{1}{\gamma\eta}\right)$$ where the number of Hamiltonian simulation steps and ancilla overhead are polylogarithmic, and the output is $\eta$-close to $|\lambda_0\rangle$ (Keen et al., 2021).

2. Spectral-Gap Amplification and Special Cases

For frustration-free Hamiltonians, a quadratic improvement is achievable by spectral-gap amplification. By replacing $H$ with a block-encoded operator $H_r$ such that $H_r^2$ mimics $H$ on the system with spectrum in $[0,1]$ but gap $\sqrt{\Delta}$, all resource estimates improve as $\Delta \to \sqrt{\Delta}$. Specifically, the overall query complexity becomes

$$O\left(\frac{1}{\gamma \sqrt{\Delta}}\log\frac{1}{\gamma\eta}\right)$$

This is realized via the Somma–Chowdhury construction, using local projectors $\Pi_j$ and ancillary registers, and is most efficiently demonstrated on models like the $q$-deformed XXZ chain, where numerical results confirm the quadratic advantage (Keen et al., 2021).

3. LCU Circuit Resources and Scalings

A detailed resource analysis for the protocol reveals:

Resource	Scaling (Generic)	Scaling (FF Hamiltonian)
LCU terms	$N = \Theta[(1/\Delta)\log(1/(\gamma\eta))]$	$N = \Theta[(1/\sqrt{\Delta})\log(1/(\gamma\eta))]$
Ancilla width	$O(\log N)$	$O(\log N + \log n)$ (due to block-encoding)
Query complexity	$O((1/(\gamma\Delta))\log(1/(\gamma\eta)))$	$O((1/(\gamma\sqrt{\Delta}))\log(1/(\gamma\eta)))$
Circuit depth	$$O(T_{max} \times \text{cost/unit Hamiltonian step})$$	$O(T_{max} \times \text{cost}/\sqrt{\Delta})$
Success probability	$\Omega(\gamma)$	Same

$$T_{max} = z_c t = \Theta\bigl(\frac{1}{\Delta}\log\frac{1}{\gamma\eta}\bigr)$$ is the maximum total simulated time. Implementations require the time-evolution oracle for $H$ to be accurate to $O(\gamma\eta/T_{max})$, and the ground energy $\lambda_0$ must be known to within $O(\Delta / \sqrt{\log(1/(\gamma\eta))})$ to shift the spectrum (Keen et al., 2021).

4. Comparison with Alternative Virtual Preparation Schemes

The LCU-based filter (virtual ground-state projection) framework is part of a broad class of virtual ground-state preparation methodologies, including:

• QSP/QSVT Filtering: Polynomial filters on $H$ constructed via QSP or QSVT, with complexity scaling as $\Delta^{-1}$ for generic Hamiltonians, and as $\Delta^{-1/2}$ for exact frustration-free cases (Thibodeau et al., 2021).
• Multi-Level Quantum Signal Processing: Utilizing repeated, logarithmically many QSP filters in cascade can exploit fast-forwarding Hamiltonian simulation to achieve complexity scaling as $O(\log(\|H\|/\Delta))$ for systems where long-time simulation is "free" (Dong et al., 4 Jun 2024).
• Qubitization and Chebyshev-LCU: Imaginary-time filtering via Chebyshev expansion of $e^{-\tau H}$, leveraging qubitization to express the action as controlled powers of a walk operator, with quadratic resource scaling in the truncation order $N$ required to reach a given error (Marteau, 2023).

These frameworks, while architecturally distinct, share the virtual feature of synthesizing ground-state projectors without genuine open-system or measurement-driven dissipation. They also illustrate optimality—no approach can beat the $\Omega(\Delta^{-1}/\gamma)$ query scaling for general local Hamiltonians (Keen et al., 2021, Thibodeau et al., 2021).

5. Algorithmic Workflow and Circuit Implementation

A step-wise outline of the LCU-based virtual ground-state projector is:

1. Spectrum Rescaling & Shifting: Normalize $H$ so $\sigma(H) \subset [0,1]$ and shift so $\lambda_0 \geq 0$.
2. LCU Construction: Compute $t$, $z_c$, $\Delta_z$, $N$, and construct weights $\alpha_k$ and times $z_k$.
3. Ancilla Preparation: Prepare $\sum_k \sqrt{\alpha_k} |k\rangle$ in the ancilla register.
4. Controlled-Time Evolution: Apply $\sum_k |k\rangle\langle k| \otimes U_k$ to implement the linear combination.
5. Ancilla Uncomputation & Post-selection: Uncompute the ancilla and measure in $|0...0\rangle$; if unsuccessful, repeat or amplitude-amplify.
6. Renormalization: Output the main register, now in a state $\epsilon$-close to $|\lambda_0\rangle$ (Keen et al., 2021).

Each U$_k = e^{-i z_k t H}$ can be realized by Trotterization, product formulas, or more advanced Hamiltonian simulation techniques.

6. Numerical Illustration and Benchmarks

Application to the $q$-deformed XXZ chain (open chain, $L=4$–$10$ sites) demonstrates:

• For $q=1$, spectral gap $\Delta_s = O(1/L^2)$; initial state overlap $\gamma \simeq (L+1)^{-1/2}$.
• Achieving infidelity $\epsilon=10^{-2}$ requires $$T_{max} \approx c\, (1/\Delta_s) \log(1/(\gamma\eta))$$, $c\approx 1\!\!-\!\!2$.
• With spectral-gap amplification: $T_{max}\sim(1/\sqrt{\Delta_s})$ for the same $\epsilon$, confirming the quadratic speedup.
• For $L=10$, circuit depth per controlled $U_k$ is $O((1/\sqrt{\Delta})L\log(1/(\gamma\eta)))$ two-qubit gates (Keen et al., 2021).

7. Significance and Theoretical Limits

The virtual ground-state preparation approach realizes a nearly optimal quantum algorithm for ground-state projection, achieving the scaling lower-bound for generic Hamiltonians. For frustration-free models, a quadratic speedup in the gap is attainable, and the method is robust to initial state choices with overlap $\gamma>0$. Major implications include:

• Near-Optimal Complexity: Query cost matches best-known lower bounds for generic and frustration-free Hamiltonians.
• Universality: Applies to arbitrary $n$-qubit Hermitian systems, given a simulable Hamiltonian and modest assumptions about the spectral gap and initial overlap.
• Versatility: Amenable to integration with QPE, variational optimizers (as overlap boosters), and more advanced techniques leveraging fast-forwarded simulation or block-encodings.
• Architecture-Agnostic Implementation: The ancilla overhead and circuit depth are compatible with early fault-tolerant quantum devices, as confirmed by explicit resource estimates and simulations (Keen et al., 2021).

In summary, the virtual ground-state preparation paradigm, particularly the LCU-based Gaussian projection filter, furnishes a foundational and scalable primitive for quantum algorithms targeting ground-state properties across quantum chemistry, condensed matter, and lattice gauge theory applications. Its theoretical guarantees, explicit circuit construction, and extensibility to enhanced scaling regimes position it as a central approach in the landscape of quantum ground-state simulation algorithms (Keen et al., 2021).