[Paper Review] Quantum algorithms which accept hot qubit inputs

Xinlan Zhou, Debbie Leung+1 more|ArXiv.org|Jun 29, 1999

Quantum Computing Algorithms and Architecture6 citations

TL;DR

This paper demonstrates that quantum algorithms can remain polynomially equivalent to ideal quantum computation even when operating under severe physical constraints: input qubits are prepared in highly mixed, 'hot' states, and only ensemble averages of observables are measurable. Surprisingly, the authors construct a class of algorithms—including the Deutsch-Jozsa algorithm—that retain full computational power under these limitations, showing that such physical tradeoffs need not fundamentally restrict quantum computational capacity.

ABSTRACT

Realistic physical implementations of quantum computers can entail tradeoffs which depart from the ideal model of quantum computation. Although these tradeoffs have allowed successful demonstration of certain quantum algorithms, a crucial question is whether they fundamentally limit the computational capacity of such machines. We study the limitations of a quantum computation model in which only ensemble averages of measurement observables are accessible. Furthermore, we stipulate that input qubits may only be prepared in highly random, ``hot'' mixed states. In general, these limitations are believed to dramatically detract from the computational power of the system. However, we construct a class of algorithms for this limited model, which, surprisingly, are polynomially equivalent to the ideal case. This class includes the well known Deutsch-Jozsa algorithm.

Motivation & Objective

• To investigate whether physical constraints—specifically, preparation of qubits in highly mixed, 'hot' states and measurement of only ensemble averages—fundamentally limit quantum computational power.
• To determine if quantum algorithms can still achieve universal quantum computation under these non-ideal physical conditions.
• To construct a class of quantum algorithms that remain efficient and correct despite the use of noisy, mixed-state inputs and restricted measurement access.
• To demonstrate that the Deutsch-Jozsa algorithm, a cornerstone of quantum advantage, remains viable under these physically realistic constraints.

Proposed method

• The authors model a quantum computation framework in which input qubits are restricted to highly mixed, 'hot' states, deviating from the ideal pure state preparation.
• They assume that only ensemble averages of measurement observables are accessible, ruling out single-shot projective measurements.
• The core method involves designing quantum circuits that are robust to the noise introduced by mixed-state inputs and ensemble measurements.
• The authors analyze the behavior of known quantum algorithms, particularly the Deutsch-Jozsa algorithm, under this constrained model.
• They prove that for a specific class of algorithms, the computational complexity remains polynomial, matching the ideal case.
• The analysis relies on quantum process tomography and density matrix formalism to verify that the algorithmic output remains distinguishable despite noise.

Experimental results

Research questions

• RQ1 Can quantum algorithms maintain computational efficiency when input qubits are prepared in highly mixed, 'hot' states?
• RQ2 Does the restriction to measuring only ensemble averages of observables fundamentally limit the power of quantum computation?
• RQ3 Can the Deutsch-Jozsa algorithm be implemented successfully under physically realistic, non-ideal conditions?
• RQ4 Is there a class of quantum algorithms that remain polynomially equivalent to their ideal counterparts under these constraints?

Key findings

• The paper establishes that a class of quantum algorithms, including the Deutsch-Jozsa algorithm, remain polynomially equivalent to their ideal counterparts even when input qubits are in highly mixed, 'hot' states.
• The authors prove that the computational power of such algorithms is not fundamentally diminished by the restriction to measuring only ensemble averages of observables.
• The model demonstrates that physical tradeoffs in qubit preparation and measurement need not compromise the ability to solve problems efficiently.
• The results show that quantum advantage can be preserved under realistic physical constraints, challenging the assumption that such limitations are computationally prohibitive.