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IEEE International Conference on Quantum Computing and Engineering (QCE)

Robustness of Quantum Algorithms Against Approximate Data Representations

Vladyslav Los, Mykola Maksymenko, Maciej Koch-Janusz, Yuriy Pryyma, Richard Givhan

Published Sep 17, 2023• 0 citations

Abstract

Quantum algorithms face practical difficulties and overheads in execution on realistic devices, potentially thwarting their theoretical advantages. A key practical challenge is efficiently encoding classical data, since data embedding layers, often exponential in the number of qubits, can consume a significant portion of the circuit depth. One possible strategy to address this is to compress the standard data embeddings using variational circuits. The approximate amplitude encoding (AAE) algorithm successfully applies this approach to financial calculations and image pattern matching, achieving only polynomially deep embedding layers. While prior works focus on optimizing the approximate embedding in isolation, the ultimate goal is the compound algorithm's performance. This study systematically examines the impact of embedding compression on the fidelity of the subsequent quantum algorithm in a noiseless setting, specifically investigating amplitude encoding and its dependence on input data set size and complexity. Trainability of the different variational ansaetze, their output state fidelity, and the degree of the embedding layer depth compression are investigated. Performance benchmarks of standard quantum algorithms, e.g. Grover search, amplitude estimation, and quantum Fourier transform (QFT), are conducted based on the degree of compression and quality of the compressed input state. Results show that the amplitude encoding circuit can be compressed substantially in depth, for generic data and a small number of qubits, with only marginal loss in state fidelity. Certain algorithms, like QFT, demonstrate robustness to such approximations.

DOI: 10.1109/qce57702.2023.10269

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Quantum algorithms face practical difficulties and overheads in execution on realistic devices, potentially thwarting their theoretical advantages. A key practical challenge is efficiently encoding classical data, since data embedding layers, often exponential in the number of qubits, can consume a significant portion of the circuit depth. One possible strategy to address this is to compress the standard data embeddings using variational circuits. The approximate amplitude encoding (AAE) algorithm successfully applies this approach to financial calculations and image pattern matching, achieving only polynomially deep embedding layers. While prior works focus on optimizing the approximate embedding in isolation, the ultimate goal is the compound algorithm's performance. This study systematically examines the impact of embedding compression on the fidelity of the subsequent quantum algorithm in a noiseless setting, specifically investigating amplitude encoding and its dependence on input data set size and complexity. Trainability of the different variational ansaetze, their output state fidelity, and the degree of the embedding layer depth compression are investigated. Performance benchmarks of standard quantum algorithms, e.g. Grover search, amplitude estimation, and quantum Fourier transform (QFT), are conducted based on the degree of compression and quality of the compressed input state. Results show that the amplitude encoding circuit can be compressed substantially in depth, for generic data and a small number of qubits, with only marginal loss in state fidelity. Certain algorithms, like QFT, demonstrate robustness to such approximations.

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Classical simulation of quantum circuits is crucial for evaluating and validating the design of new quantum algorithms. However, the number of quantum state amplitudes increases exponentially with the number of qubits, leading to the exponential growth of the memory requirement for the simulations. In this paper, we present a new data reduction technique to reduce the memory requirement of quantum circuit simulations. We apply our amplitude-aware lossy compression technique to the quantum state amplitude vector to trade the computation time and fidelity for memory space. The experimental results show that our simulator only needs 1/16 of the original memory requirement to simulate Quantum Fourier Transform circuits with 99.95% fidelity. The reduction amount of memory requirement suggests that we could increase 4 qubits in the quantum circuit simulation comparing to the simulation without our technique. Additionally, for some specific circuits, like Grover's search, we could increase the simulation size by 18 qubits.

Quantum state preparation (QSP) for a general

n

-qubit state requires

O(2^n)

CNOT gates and circuit depth, making exact amplitude encoding (EAE) impractical for near-term quantum hardware. We introduce an ancilla-free hybrid classical-quantum strategy that reduces this cost to

O(poly(n))

for a broad class of compressible data. The method first applies a classical compression step to obtain a

d

-sparse representation of the input, loads this sparse vector using a sparse-state preparation routine, and then reconstructs the target state through a polynomial-depth quantum inverse transform. We evaluate the framework on synthetic benchmark signals and real biomedical time series using Fourier and Haar transforms, demonstrating substantial reductions in CNOT counts and circuit depth compared to EAE, together with competitive performance relative to the Fourier Series Loader (FSL). The quantum simulation results show that combining classical data compression with quantum decompression provides a scalable framework for efficient QSP, reducing quantum overhead without requiring variational training or ancillary registers.

Amplitude embedding (AE) is essential in quantum machine learning (QML) for encoding classical data onto quantum circuits. However, conventional AE methods suffer from deep, variable-length circuits that introduce high output error due to extensive gate usage and variable error rates across samples, resulting in noise-driven inconsistencies that degrade model accuracy. We introduce EnQode, a fast AE technique based on symbolic representation that addresses these limitations by clustering dataset samples and solving for cluster mean states through a low-depth, machine-specific ansatz. Optimized to reduce physical gates and SWAP operations, EnQode ensures all samples face consistent, low noise levels by standardizing circuit depth and composition. With over 94% fidelity in data mapping, EnQode enables robust, high-performance QML on noisy intermediate-scale quantum (NISQ) devices. Our opensource solution provides a scalable and efficient alternative for integrating classical data with quantum models.

We propose using variational quantum algorithms (VQAs) to simulate established quantum algorithms under realistic noise conditions, aiming to surpass the fidelity of theoretical circuits in noisy environments. Focusing on the Quantum Fourier Transform (QFT), we perform numerical simulations for two qubits under both coherent and incoherent noise. To enhance generalization, we further introduce the use of Mutually Unbiased Bases (MUBs) during the optimization. Our results show that the variational circuit can reproduce the QFT with higher fidelity in scenarios dominated by coherent noise. This demonstrates the potential of the approach as an effective error-mitigation strategy for small- to medium-scale quantum systems, particularly in settings where coherent noise strongly impacts performance. Beyond mitigating noise and improving fidelity, the method can be adapted to the noise profile of a specific device, providing a versatile and practical route to enhance the reliability of quantum algorithms in near-term quantum hardware.

Amplitude encoding of real-world data on quantum computers is often the workflow bottleneck: direct amplitude encoding scales poorly with input size and can offset any speedups in subsequent processing. Fourier-based sparse amplitude encoding lowers cost by retaining only a small subset of dominant coefficients, but its fixed, non-adaptive basis leads to significant information loss. In this work, we replace the Fourier transform with the adaptive interpolating quantum transform (AIQT) in the sparse amplitude encoding workflow. The AIQT learns a data-adapted basis that concentrates information into a small number of coefficients. Consequently, at matched sparsity, the AIQT retains more information and achieves lower reconstruction error compared to the Fourier baseline. On financial time-series data, the AIQT reduces reconstruction error by 40% relative to the Fourier baseline, and on image datasets the reduction is up to 50% at the same sparsity level, with nearly identical encoding gate cost. Crucially, the approach preserves the efficiency of Fourier-based methods: the AIQT is built on the structure of the quantum Fourier transform circuit. Its gate count scales quadratically with the number of qubits, while classical evaluation can be carried out in quasilinear time. In addition, the AIQT is trained without labels and does not require sampling from quantum hardware or a simulator, removing a major bottleneck in data-driven amplitude-encoding methods.