Efficient quantum state preparation on Quantinuum hardware

Technical Summary: Efficient Quantum State Preparation on Quantinuum Hardware

Problem Statement
Accurate quantum state preparation and subsequent fidelity verification are fundamental prerequisites for realizing quantum advantages in applications such as quantum image processing, machine learning, and chemistry. However, current experimental realizations face two primary bottlenecks:

1. Resource Efficiency: Preparing complex, structured target states (e.g., encoding classical data) often requires deep circuits or excessive resources, which are prohibitive on Near-Term Intermediate Scale Quantum (NISQ) devices.
2. Verification Scalability: Validating the fidelity of a prepared state is challenging. While "shadow overlap" methods (Huang et al.) offer a way to certify fidelity using local measurements, their effectiveness is governed by a parameter $\tau$ (the relaxation time of a Markov chain associated with the target state). For highly structured states common in practical algorithms, $\tau$ can become pathologically large, rendering the shadow overlap method inefficient or ineffective due to prohibitively high sample complexity.

Methodology
The authors propose an end-to-end framework combining resource-efficient state preparation with a novel verification technique, implemented on the Quantinuum H2-1 trapped-ion platform.

1. State Preparation via Tensor Networks:

   • The target state is a digitized acoustic signal (a segment of a Gaussian-modulated tone) encoded into a 13-qubit system ($N=8192$ samples).
   • The preparation utilizes Schmidt Spectrum Optimisation (SSO), a tensor-network-based strategy. This algorithm constructs a shallow quantum circuit by classically optimizing a sequence of local unitaries (staircase layers of $SU(4)$ gates) to disentangle the target state.
   • The process is reversed to generate the preparation circuit: a low-rank Matrix Product State (MPS) is prepared exactly, followed by the application of the inverse unitaries. This approach minimizes circuit depth and avoids ancilla qubits.

2. $\tau$-Reducing Layer (Basis Change):

   • To address the limitation of shadow overlap certification for structured states, the authors introduce a pre-measurement basis-change layer.
   • This layer consists of single-qubit gates optimized to minimize the Inverse Participation Ratio (IPR) of the state amplitudes.
   • Theoretically, minimizing IPR pushes the probability distribution toward uniformity, which correlates with reducing the relaxation time $\tau$ of the associated Markov chain.
   • Crucially, this layer can be computed efficiently for MPS targets and absorbed into the final layer of the preparation circuit, adding no additional circuit depth.

3. Shadow Overlap Certification:

   • The prepared state is verified using the shadow overlap method. The $\tau$-reducing layer is applied prior to measurement to ensure the parameter $\tau$ remains within a tractable range, tightening the theoretical bounds on the relationship between the observed shadow overlap and the true fidelity.

Key Results

• Experimental Implementation: The framework was executed on the Quantinuum H2-1 device using a circuit of 229 single-qubit and 108 two-qubit gates.
• Fidelity Achievements:
  • The classical SSO algorithm achieved an approximation fidelity of $F_{SSO} = 0.9958756$ between the ideal circuit target and the original acoustic signal.
  • The hardware preparation fidelity relative to the ideal circuit target was estimated as $F_{hw} = 0.929$.
  • The shadow overlap measured on the hardware was $\bar{\omega}_{H2-1} = 0.967$, compared to $0.971$ on the noisy emulator and $0.984$ in ideal simulation.
• $\tau$ Reduction:
  • For the acoustic signal target, the basis change reduced $\tau$ from 447.29 to 17.49.
  • Numerical experiments on other structured states (sinusoidal wave packets, 1D MBL spin chains, 1D Hubbard chains, and 2D Heisenberg models) demonstrated that the $\tau$-reducing layer consistently lowers $\tau$ by over 10 orders of magnitude, bringing it down to the order of $O(n)$.
• Efficiency: The entire verification procedure was completed with just 1,000 measurement shots, demonstrating a scalable benchmarking standard.

Significance and Claims
The paper claims to shift the paradigm for mapping and verifying structured classical data on quantum hardware under realistic noise and measurement constraints. Its primary contributions are:

1. End-to-End Workflow: A unified integration of tensor-network preparation and shadow validation that operates effectively on NISQ-era and early fault-tolerant devices without relying on idealized assumptions or deep fault-tolerant overhead.
2. Mitigation of Certification Limits: By introducing the $\tau$-reducing layer, the work overcomes a key theoretical limitation of the shadow overlap method, making it viable for structured states that previously exhibited pathologically high $\tau$ values.
3. Practical Robustness: The demonstration of high-fidelity state preparation ($F_{hw} \approx 0.93$) and rapid verification on a 13-qubit system provides an immediate, scalable standard for benchmarking quantum devices.
4. Theoretical Implication: The results suggest that the shadow overlap method can be an efficient certification procedure for a broad range of structured states if the $\tau$-reducing layer is employed, potentially resolving open questions regarding the scalability of shadow-based certification.

The authors note that future work could extend this scheme to larger medical imaging datasets, leveraging improvements in quantum hardware, while theoretically establishing analytically that the $\tau$-reducing layer can universally bring $\tau$ down to $O(n)$.