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arXiv:2107.14127 · Physical Review Research

Quantum state preparation protocol for encoding classical data into the amplitudes of a quantum information processing register's wave function

Sahel Ashhab

Published Feb 3, 2022• 30 citations
arXivHTMLPDF

Abstract

We present a protocol for encoding $N$ real numbers stored in $N$ memory registers into the amplitudes of the quantum superposition that describes the state of ${log}_{2}N$ qubits. This task is one of the main steps in quantum machine learning algorithms applied to classical data. The protocol combines partial CNOT gate rotations with probabilistic projection onto the desired state. The number of additional ancilla qubits used during the implementation of the protocol, as well as the number of quantum gates, scales linearly with the number of qubits in the processing register and hence logarithmically with $N$. The average time needed to successfully perform the encoding scales logarithmically with the number of qubits, in addition to being inversely proportional to the acceptable error in the encoded amplitudes. It also depends on the structure of the dataset in such a way that the protocol is most efficient for nonsparse data.

DOI
10.1103/physrevresearch.4.013091

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Quantum state preparation, also known as encoding or embedding, is a crucial initial step in many quantum algorithms and often constrains theoretical quantum speedup in fields such as quantum machine learning and linear equation solvers. One common strategy is amplitude encoding, which embeds a classical input vector of sizeN=2n\mathrm{N}=2^{\mathrm{n}}N=2nin the amplitudes of an n-qubit register. For arbitrary vectors, the circuit depth typically scales linearly with the input size N, rapidly becoming unfeasible on near-term hardware. We propose a general-purpose procedure with sublinear average depth in N, increasing the window of utility. Our amplitude encoding method encodes arbitrary complex vectors of sizeN=2n\mathrm{N}=2^{n}N=2nat any desired binary precision using a register withnnnqubits plus 2 ancillas and a sublinear number of multi-controlled NOT (MCX) gates, at the cost of a probabilistic success rate proportional to the sparsity of the encoded data. The core idea of our procedure is to construct an isomorphism between target states and hypercube graphs, in which specific reflections correspond to MCX gates. This reformulates the state preparation problem in terms of permutations and binary addition. The use of MCX gates as fundamental operations makes this approach particularly suitable for quantum platforms such as ion traps and neutral atom devices. This geometrical perspective paves the way for more gate-efficient algorithms suitable for nearterm hardware applications.
Quantum state preparation (QSP) for a general nnn-qubit state requires O(2n)O(2^n)O(2n) 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))O(poly(n))O(poly(n)) for a broad class of compressible data. The method first applies a classical compression step to obtain a ddd-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.
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Classical data could be transformed into quantum states through quantum amplitude encodings with exponential compression, but existing protocols suffer from a sharp drop in success probability when the data is sparse or has a large dynamic range. In this work, we formulate a novel encoding algorithm which requires a minimal amount of ancillary qubits. By employing a three-step quantum circuit consisting of matching, rotation, and flipping, we boost the minimum success probability to 1 − 2−twith t ancillary qubits, independent of the data distribution. Theoretical analysis and circuit design demonstrate that this approach maintains logarithmic-scale resource consumption while being universally applicable to arbitrary datasets. This provides an efficient and robust data-loading frontend for quantum machine learning and linear algebra applications in the NISQ era.
Quantum state preparation is a central primitive in many quantum algorithms, yet it is generally resource intensive, with efficient constructions known only for structured families of states. This work introduces a method for preparing quantum states whose amplitudes are given by a degree−d-d−d polynomial, using circuits with logarithmic depth in the number nnn of qubits and only O(n)\mathcal O(n)O(n) ancilla qubits, improving previous approaches that required linear-depth circuits. The construction first relies on a block-encoding of an affine diagonal operator based on its Pauli-basis decomposition, which involves only nnn terms. A modified linear-combination-of-unitaries (LCU) technique is introduced to implement this decomposition in logarithmic depth, together with a novel circuit for the EXACT-one oracle that flags basis states in which exactly one qubit is in the state ∣1⟩|1\rangle∣1⟩. It then uses a generalized quantum eigenvalue transformation (GQET) to promote this affine operator to an arbitrary degree polynomial. Theoretical analysis and numerical simulations are reported along with a proof-of-principle implementation on a trapped-ion quantum processor using 141414 qubits and more than 500500500 primitive quantum gates. Because polynomial approximations are ubiquitous in scientific computing, this construction provides a scalable and resource-efficient approach to quantum state preparation, further improving the potential of quantum algorithms in fields such as chemistry, physics, engineering, and finance.
A prerequisite for many quantum information processing tasks to truly surpass classical approaches is an efficient procedure to encode classical data in quantum superposition states. In this work, we present a circuit-based flip-flop quantum random access memory to construct a quantum database of classical information in a systematic and flexible way. For registering or updating classical data consisting of M entries, each represented by n bits, the method requires O(n) qubits and O(Mn) steps. With post-selection at an additional cost, our method can also store continuous data as probability amplitudes. As an example, we present a procedure to convert classical training data for a quantum supervised learning algorithm to a quantum state. Further improvements can be achieved by reducing the number of state preparation queries with the introduction of quantum forking.