[Paper Review] Ideal state discrimination with an O(1)-qubit quantum computer

Robin Blume-Kohout, Sarah Croke+1 more|arXiv (Cornell University)|Jan 31, 2012

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a method for optimal quantum state discrimination between K nonorthogonal pure states using a one-at-a-time coherent measurement protocol with an O(1)-qubit quantum processor. By sequentially entangling the processor with each of N copies of a state via unitary interactions, all information about the unknown state is coherently transferred into the processor’s state, enabling optimal discrimination without joint measurements or decoherence, requiring only log₂K qubits of quantum memory.

ABSTRACT

We show how to optimally discriminate between K distinct quantum states, of which N copies are available, using one-at-a-time interactions with each of the N copies. While this task (famously) requires joint measurements on all N copies, we show that it can be solved with one-at-a-time "coherent measurements" performed by an apparatus with log(K) qubits of quantum memory. We apply the same technique to optimal discrimination between K distinct N-particle matrix product states of bond dimension D, using a coherent measurement apparatus with log(K) + log(D) qubits of memory.

Motivation & Objective

• To address the challenge of optimal quantum state discrimination between K nonorthogonal states when N copies are available, which classically requires joint measurements.
• To overcome the limitation of one-at-a-time measurements failing to achieve optimal success probability in standard quantum measurement schemes.
• To demonstrate that optimal discrimination can be achieved using a small quantum processor with O(1) qubits by replacing decoherence-based measurements with coherent unitary interactions.
• To extend the method to matrix product states (MPS) with bond dimension D, showing that log₂K + log₂D qubits suffice for optimal discrimination.

Proposed method

• Use a quantum processor (QIP) with log₂K qubits as a coherent measurement apparatus, initialized in |0⟩, to sequentially interact with each of the N system copies.
• Implement a SWAP operation between the first system and the QIP to transfer the state information coherently into the processor’s Hilbert space.
• For subsequent systems, apply a unitary interaction that maps the joint state of the system and processor into a product state where the processor encodes the full information about the unknown state.
• Use Gram-Schmidt orthogonalization to construct a K-dimensional basis for the subspace spanned by the N-copy states, enabling unitary rotation into the processor’s state space.
• Postpone all measurement and readout until the end, preserving coherence and enabling optimal discrimination through unitary evolution alone.
• Extend the protocol to matrix product states (MPS) by accounting for bond dimension D, requiring log₂K + log₂D qubits to coherently encode the full state information.

Experimental results

Research questions

• RQ1 Can optimal discrimination between K nonorthogonal pure states be achieved using only one-at-a-time interactions with a small quantum processor, avoiding joint measurements?
• RQ2 Is it possible to achieve optimal success probability in state discrimination without decohering the measurement apparatus, by preserving quantum coherence throughout the process?
• RQ3 How does the required quantum memory size scale with the number of candidate states K and the complexity of the state structure, such as in matrix product states?
• RQ4 Can the coherent measurement protocol be generalized to many-body entangled states like MPS with finite bond dimension D?
• RQ5 What is the fundamental role of coherence in measurement, and can unitary evolution alone serve as a complete information-gathering process?

Key findings

• Optimal discrimination between K nonorthogonal pure states using N copies can be achieved with a quantum processor of only log₂K qubits, using one-at-a-time coherent interactions.
• The protocol avoids decoherence and wavefunction collapse, relying solely on unitary evolution to transfer and preserve information about the unknown state.
• For matrix product states (MPS) with bond dimension D, optimal discrimination requires a processor of size log₂K + log₂D qubits, demonstrating scalability to many-body entangled states.
• The method achieves the same optimal success probability as joint measurements, but without requiring entangled operations across all N copies.
• The protocol is robust to the number of copies N and only requires sequential, local interactions with each system, making it feasible for near-term quantum devices.
• The results suggest that coherent measurements—unitary interactions without decoherence—can outperform standard measurements in information gain, challenging the conventional view of measurement as inherently irreversible.