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A direct approach to fault-tolerance in measurement-based quantum computation via teleportation

Marcus Palmer Da Silva, Vincent Danos A, Elham Kashefi, Ollivier, Harold

Published Jun 1, 2007• 10 citations
OpenAlex

Abstract

We discuss a simple variant of the one-way quantum computing model [R. Raussendorf and H.-J. Briegel, PRL 86, 5188, 2001], called the Pauli measurement model, where measurements are restricted to be along the eigenbases of the Pauli X and Y operators, while auxiliary qubits can be prepared both in the $\ket{+_{\pi\over 4}}:={1/\sqrt{2}}(\ket{0}+e^{i{\pi\over 4}}\ket{1})$ state, and the usual $\ket{+}:={1/ \sqrt{2}}(\ket{0}+\ket{1})$ state. We prove the universality of this quantum computation model, and establish a standardization procedure which permits all entanglement and state preparation to be performed at the beginning of computation. This leads us to develop a direct approach to fault-tolerance by simple transformations of the entanglement graph and preparation operations, while error correction is performed naturally via syndrome-extracting teleportations.

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We discuss a simple variant of the one-way quantum computing model (Raussendorf R and Briegel H-J 2001 Phys. Rev. Lett. 86 5188), called the Pauli measurement model, where measurements are restricted to be along the eigenbases of the Pauli X and Y operators, while qubits can be initially prepared both in the state and the usual state. We prove the universality of this quantum computation model, and establish a standardization procedure which permits all entanglement and state preparation to be performed at the beginning of computation. This leads us to develop a direct approach to fault-tolerance by simple transformations of the entanglement graph and preparation operations, while error correction is performed naturally via syndrome-extracting teleportations.

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Introduction In the standard model of quantum computation [D95], some simple quantum state is prepared and evolved by a partially ordered set of unitary gates, and finally is measured in the computation basis. Up until the late 1990s, transformations that deviate from unitarity were commonly associated with irreversibility, and thus incoherence and harm induced in the quantum data of interest. However, this viewpoint was challenged by three notable results: (1) teleportation [BBC+93], (2) syndrome measurements [S95, S96e], and (3) the fault-tolerant Toffoli and π/8-gate constructions [S96b, BMP+99]. In these schemes, well-designed measurements project the quantum data of interest onto a subspace of the original ambient space for each outcome, up to some known unitary transformation that is either reversible (and harmless) or intended. Subsequently, researchers found many useful error-correction procedures and unitary gates that rely heavily on measurements. Two “measurement models” of quantum computation were proposed in the early 2000s that make extreme use of measurements. In the cluster model [RB01], any quantum computation can be performed by applying a partially time-ordered set of single-qubit measurements on a certain entangled state. The state must be of appropriate size, but is otherwise independent of the computation. The related graph state model [RBB03] uses an initial state that depends on the computation itself. In the seemingly different teleportation-based model [N03, L01a, L04], any quantum computation can be implemented by applying one- or two-qubit nondemolition measurements on an arbitrary initial state, so as to “teleport” each gate in the simulated circuit.

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Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer [Raussendorf and Briegel 2001] stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns , and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models, for example, Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.

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Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computater was introduced by Briegel and Raussendorf and quantum computation via projective measurements only by Nielsen. The more recent development of this second model is based on state transfers instead of teleportation. From this development a finite but approximate quantum universal family of observables is exhibited which includes only one two-qubit observable while others are one-qubit observables. In this article an infinite but exact quantum universal family of observables is proposed including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison which was initiated by Cirac and Verstraete closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation.

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Unifying Quantum Computation with Projective Measurements only and One-Way Quantum Computation

Philippe Jorrand, Perdrix, S

Abstract. Quantum measurement is universal for quantum computation (Nielsen [4], Raussendorf [7,8]). Two models for performing measurement-based quantum computation exist: the one-way quantum computer was introduced by Briegel and Raussendorf [7], and quantum computation via projective measurements only by Nielsen [4]. The more recent development of this second model is based on state transfers [6] instead of teleportation. From this development, a finite but approximate quantum universal family of observables is exhibited, which includes only one two-qubit observable, while others are one-qubit observables [6]. In this article, an infinite but exact quantum universal family of observables is proposed, including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation, i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison, which was initiated by Cirac and Verstraete [9], closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation. 1

•Jan 1, 2004•29 citations
We discuss a simple variant of the one-way quantum computing model [R. Raussendorf and H.-J. Briegel, PRL 86, 5188, 2001], called the Pauli measurement model, where measurements are restricted to be along the eigenbases of the Pauli X and Y operators, while auxiliary qubits can be prepared both in the ∣+π4⟩:=1/2(∣0⟩+eiπ4∣1⟩)\ket{+_{\pi\over 4}}:={1/\sqrt{2}}(\ket{0}+e^{i{\pi\over 4}}\ket{1})∣+4π​​⟩:=1/2​(∣0⟩+ei4π​∣1⟩) state, and the usual ∣+⟩:=1/2(∣0⟩+∣1⟩)\ket{+}:={1/ \sqrt{2}}(\ket{0}+\ket{1})∣+⟩:=1/2​(∣0⟩+∣1⟩) state. We prove the universality of this quantum computation model, and establish a standardization procedure which permits all entanglement and state preparation to be performed at the beginning of computation. This leads us to develop a direct approach to fault-tolerance by simple transformations of the entanglement graph and preparation operations, while error correction is performed naturally via syndrome-extracting teleportations.
We discuss a simple variant of the one-way quantum computing model (Raussendorf R and Briegel H-J 2001 Phys. Rev. Lett. 86 5188), called the Pauli measurement model, where measurements are restricted to be along the eigenbases of the Pauli X and Y operators, while qubits can be initially prepared both in the state and the usual state. We prove the universality of this quantum computation model, and establish a standardization procedure which permits all entanglement and state preparation to be performed at the beginning of computation. This leads us to develop a direct approach to fault-tolerance by simple transformations of the entanglement graph and preparation operations, while error correction is performed naturally via syndrome-extracting teleportations.
Introduction In the standard model of quantum computation [D95], some simple quantum state is prepared and evolved by a partially ordered set of unitary gates, and finally is measured in the computation basis. Up until the late 1990s, transformations that deviate from unitarity were commonly associated with irreversibility, and thus incoherence and harm induced in the quantum data of interest. However, this viewpoint was challenged by three notable results: (1) teleportation [BBC+93], (2) syndrome measurements [S95, S96e], and (3) the fault-tolerant Toffoli and π/8-gate constructions [S96b, BMP+99]. In these schemes, well-designed measurements project the quantum data of interest onto a subspace of the original ambient space for each outcome, up to some known unitary transformation that is either reversible (and harmless) or intended. Subsequently, researchers found many useful error-correction procedures and unitary gates that rely heavily on measurements. Two “measurement models” of quantum computation were proposed in the early 2000s that make extreme use of measurements. In the cluster model [RB01], any quantum computation can be performed by applying a partially time-ordered set of single-qubit measurements on a certain entangled state. The state must be of appropriate size, but is otherwise independent of the computation. The related graph state model [RBB03] uses an initial state that depends on the computation itself. In the seemingly different teleportation-based model [N03, L01a, L04], any quantum computation can be implemented by applying one- or two-qubit nondemolition measurements on an arbitrary initial state, so as to “teleport” each gate in the simulated circuit.
Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer [Raussendorf and Briegel 2001] stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns , and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models, for example, Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.
Quantum computation provides us with a new form of information processing that surpasses current computer technology.In a new model of quantum computation, measurement-based quantum computation, the computation proceeds via simple measurements on previously prepared quantum entangled states.This model has attracted attention for its good potential for realizing quantum computers.This article reviews the basic concepts of measurement-based quantum computation and the elegant fault-tolerant system called topological one-way quantum computation.
Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computater was introduced by Briegel and Raussendorf and quantum computation via projective measurements only by Nielsen. The more recent development of this second model is based on state transfers instead of teleportation. From this development a finite but approximate quantum universal family of observables is exhibited which includes only one two-qubit observable while others are one-qubit observables. In this article an infinite but exact quantum universal family of observables is proposed including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison which was initiated by Cirac and Verstraete closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation.
Abstract. Quantum measurement is universal for quantum computation (Nielsen [4], Raussendorf [7,8]). Two models for performing measurement-based quantum computation exist: the one-way quantum computer was introduced by Briegel and Raussendorf [7], and quantum computation via projective measurements only by Nielsen [4]. The more recent development of this second model is based on state transfers [6] instead of teleportation. From this development, a finite but approximate quantum universal family of observables is exhibited, which includes only one two-qubit observable, while others are one-qubit observables [6]. In this article, an infinite but exact quantum universal family of observables is proposed, including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation, i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison, which was initiated by Cirac and Verstraete [9], closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation. 1