[Paper Review] An introduction to measurement based quantum computation

Richard Jozsa|ArXiv.org|Aug 17, 2005

Quantum Information and Cryptography23 references123 citations

TL;DR

This paper introduces measurement-based quantum computation (MBQC), where universal quantum computation is achieved through adaptive measurements on a fixed entangled resource state—specifically, the teleportation-based quantum computing (TQC) and one-way quantum computer (1WQC) models. The key contribution is demonstrating that MBQC enables natural parallelization of quantum algorithms and offers a structural separation between classical and quantum processing layers, suggesting potential exponential reductions in quantum depth for polynomial-time algorithms.

ABSTRACT

In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The choice of basis for later measurements may depend on earlier measurement outcomes and the final result of the computation is determined from the classical data of all the measurement outcomes. This is in contrast to the more familiar gate array model in which computational steps are unitary operations, developing a large entangled state prior to some final measurements for the output. Two principal schemes of measurement based computation are teleportation quantum computation (TQC) and the so-called cluster model or one-way quantum computer (1WQC). We will describe these schemes and show how they are able to perform universal quantum computation. We will outline various possible relationships between the models which serve to clarify their workings. We will also discuss possible novel computational benefits of the measurement based models compared to the gate array model, especially issues of parallelisability of algorithms.

Motivation & Objective

• To present measurement-based quantum computation as a viable alternative to the gate array model.
• To explain how universal quantum computation can be achieved using only measurements on a fixed entangled state.
• To explore the structural advantages of MBQC, particularly in enabling parallelization and separating classical and quantum processing layers.
• To investigate the potential computational benefits of MBQC over the gate array model, especially in terms of algorithmic depth and fault tolerance.

Proposed method

• Using the teleportation-based quantum computing (TQC) model, where quantum gates are implemented via entangled Bell-state measurements and basis rotations.
• Employing the one-way quantum computer (1WQC) model, which uses a cluster state as a universal resource and performs computation via single-qubit measurements in adaptive bases.
• Applying the concept of 'rotated Bell bases' to implement arbitrary single-qubit gates via measurement-induced teleportation.
• Using the mathematical projection formalism (Lemma 1) to show that projecting onto a maximally entangled state teleports a state with a unitary correction.
• Demonstrating that any quantum circuit can be mapped to a measurement pattern with at most linear resource overhead in the number of gates.
• Introducing a layered formalism where quantum layers (depth-1) are interspersed with classical computation layers, enabling structural analysis of quantum algorithms.

Experimental results

Research questions

• RQ1 Can universal quantum computation be achieved using only measurements, without unitary evolution as the primary computational step?
• RQ2 How do the TQC and 1WQC models relate to each other, and what are their respective computational advantages?
• RQ3 Can measurement-based models naturally support parallelization of quantum algorithms, especially when the gate array model requires sequential execution?
• RQ4 What is the relationship between classical post-processing and quantum operations in algorithms like Shor’s, and can this be formalized in MBQC?
• RQ5 Is it possible to implement any polynomial-time quantum algorithm using only O(log n) quantum layers, with classical computation in between?

Key findings

• The TQC and 1WQC models are both capable of universal quantum computation using only measurements on a fixed entangled resource state.
• Adaptive measurements in MBQC can simulate any quantum circuit with at most a linear resource overhead in the number of gates.
• Measurements on spatially separated qubits in an entangled state commute when the measurement basis is non-adaptive, enabling natural parallelization of quantum operations.
• The formalism of MBQC naturally separates classical and quantum processing, with quantum layers being depth-1 and classical layers handling basis choices based on measurement outcomes.
• For Shor’s algorithm, Cleve and Watrous have shown that the conjecture of O(log n) quantum layers holds, supporting the possibility of exponential reduction in quantum depth.
• The measurement-based model offers a new structural perspective on quantum algorithms, suggesting that polynomial-time quantum computation may require minimal quantum depth when classical processing is used effectively.