[Paper Review] Encoding classical data into a quantum computer

Kumar Ghosh|arXiv (Cornell University)|Jul 16, 2021

Quantum Computing Algorithms and Architecture19 references12 citations

TL;DR

This paper proposes a method to efficiently encode classical data into a quantum state using Schmidt decomposition and singular value decomposition (SVD), reducing the number of two-qubit gates required from O(2^{2n}) to O(2^n) for an n-qudit system. The approach constructs quantum circuits that prepare arbitrary multi-qubit states with fewer entangling gates, significantly lowering circuit complexity and error rates in quantum computation.

ABSTRACT

In this article we describe a technique to transfer data from classical domain to quantum domain. We consider a set of $N (=2^n)$ classical data in the form of a column matrix and prepare a $n$-qubit quantum state, whose components correspond to the $N$ classical data. To prepare this $n$-qubit quantum vector we use Schmidt decomposition and singular value decomposition techniques respectively and construct the corresponding family of quantum circuits. To strengthen our argument we also give specific examples by considering a set of 4 and 16 classical data and constructing the corresponding 2 and 4-qubit quantum vector respectively.

Motivation & Objective

• Address the challenge of efficiently transferring classical data into quantum states for use in quantum algorithms.
• Reduce the number of two-qubit quantum gates required to prepare arbitrary n-qubit states, which is critical for minimizing error rates in near-term quantum devices.
• Develop a scalable quantum circuit construction method based on Schmidt decomposition and SVD to minimize gate count and circuit depth.
• Demonstrate the method with explicit examples using 4 and 16 classical data points encoded into 2- and 4-qubit quantum states.
• Provide a framework for constructing quantum circuits that can be generalized to odd-numbered qubit systems as well.

Proposed method

• Use singular value decomposition (SVD) to decompose the classical data matrix into unitary matrices U, Σ, and V†, enabling efficient state preparation.
• Apply Schmidt decomposition to express the target quantum state as a sum of entangled basis states with Schmidt coefficients λ_i, derived from the singular values.
• Construct a quantum circuit in four steps: prepare Schmidt coefficients on the first n qubits via single-qubit rotations, apply CNOT gates to entangle the first and second n-qubit registers, perform basis transformations using Pauli-X and SWAP gates, and apply rotations to the second register to implement the Schmidt basis.
• Leverage the unitarity of transformation matrices to ensure the basis changes are physically realizable via quantum gates, though finding optimal gate sequences remains challenging.
• Use rotation gates U(θ) with angles derived from singular values (e.g., θ = arcsin(σ_i)) to prepare the Schmidt coefficients on individual qubits.
• Generalize the method to odd-numbered qubit systems by splitting the system into (n) and (n+1)-qubit subsystems and applying SVD to the resulting matrix.

Experimental results

Research questions

• RQ1 How can classical data be encoded into a multi-qubit quantum state with minimal two-qubit gate count?
• RQ2 Can Schmidt decomposition and SVD be used to reduce the number of entangling gates required for preparing arbitrary quantum states?
• RQ3 What is the gate complexity of preparing a 4-qubit quantum state using this method compared to standard approaches?
• RQ4 How can basis transformations (e.g., from computational to Schmidt basis) be implemented efficiently in a quantum circuit?
• RQ5 What are the practical limitations in implementing this method, particularly in terms of classical pre-processing and gate synthesis?

Key findings

• The method reduces the number of two-qubit gates required to prepare an n-qubit state from O(2^{2n}) to O(2^n), representing a significant exponential reduction in circuit depth and error probability.
• For a 4-qubit state encoding 16 classical data points, the circuit uses only 2 CNOT gates and 4 single-qubit rotation gates, with basis transformations implemented via Pauli-X and SWAP gates.
• The Schmidt coefficients are derived directly from the singular values of the data matrix, ensuring exact state preparation when SVD is performed accurately.
• The basis transformation step, though complex, is feasible using known quantum gate decompositions, with the number of two-qubit gates bounded by O(2^n) for n-qubit systems.
• The method is generalizable to odd-numbered qubit systems by splitting the register into unequal parts and applying SVD to the resulting matrix.
• The approach enables efficient preparation of arbitrary quantum states, which can then be used as input for more complex quantum algorithms such as quantum machine learning or Shor’s algorithm.