We first consider various methods for the indirect implementation of unitary gates. We apply these methods to rederive the universality of 4-qubit measurements based on a scheme much simpler than Nielsen's original construction [quant-ph/0108020]. Then, we prove the universality of simple discrete sets of 2-qubit measurements, again using a scheme simplifying the initial construction [quant-ph/0111122]. Finally, we show how to use a single 4-qubit measurement to achieve universal quantum computation, and outline a proof for the universality of almost all maximally entangling 4-qubit measurements.
We show that the Knill Laflamme Milburn method of quantum computation with linear optics gates can be interpreted as a one-way, measurement based quantum computation of the type introduced by Briegel and Rausendorf. We also show that the permanent state of n n-dimensional systems is a universal state for quantum computation.
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 fo…
Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computer 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 …
In measurement-based quantum computation, quantum algorithms are implemented via sequences of measurements. We describe a translationally invariant finite-range interaction on a one-dimensional qudit chain and prove that a single-shot measurement of the energy of an appropriate computational basis state with respect to this Hamiltonian provides the output of any quantum circuit. The required measurement accuracy scales inverse polynomially with the size of the simulated quantum circuit. This shows that the implementation of energy measurements on generic qudit chains is as hard as the realizat…
Measurement-based quantum computation is a framework of quantum computation, where entanglement is used as a resource and local measurements on qubits are used to drive the computation. It originates from the one-way quantum computer of Raussendorf and Briegel, who introduced the so-called cluster state as the underlying entangled resource state and showed that any quantum circuit could be executed by performing only local measurement on individual qubits. The randomness in the measurement outcomes can be dealt with by adapting future measurement axes so that computation is deterministic. Subs…
We discuss the interdependence of resource state, measurement setting and temporal order in measurement-based quantum computation. The possible temporal orders of measurement events are constrained by the principle that the randomness inherent in quantum measurement should not affect the outcome of the computation. We provide a classification for all temporal relations among measurement events compatible with a given initial stabilizer state and measurement setting, in terms of a matroid. Conversely, we show that classical processing relations necessary for turning the local measurement outcom…
We improve the upper bound on the minimal resources required for measurement-based quantum computation. Minimizing the resources required for this model is a key issue for experimental realization of a quantum computer based on projective measurements. This new upper bound allows also to reply in the negative to the open question about the existence of a trade-off between observable and ancillary qubits in measurement-based quantum computation.
We report first steps towards elucidating the relationship between contextuality, measurement-based quantum computation (MBQC) and the non-classical logic of a topos associated with the computation. We show that, in a class of MBQC, classical universality requires non-classical logic, which is 'consumed' during the course of the computation, thereby pinpointing another potential quantum computational resource.
We show that the usefulness of the thermal state of a specific spin-lattice model for measurement-based quantum computing exhibits a transition between two distinct "phases" - one in which every state is a universal resource for quantum computation, and another in which any local measurement sequence can be simulated efficiently on a classical computer. Remarkably, this transition in computational power does not coincide with any phase transition, classical or quantum, in the underlying spin-lattice model.
We show that at least 2kTln2 of heat dissipation per qubit occurs in measurement-based quantum computation according to Landauer's principle. This result is derived by using only the fundamental fact that quantum physics respects the no-signaling principle.
Quantum measurement is a class of quantum channels that sends quantum states to classical states. We set up resource theories of quantum coherence and quantum entanglement for quantum measurements and find relations between them. For this, we conceive a relative entropy type quantity to account for the quantum resources of quantum measurements. The quantum coherence of a quantum measurement can be converted into the entanglement in a bipartite quantum measurement through coherence-non generating transformations. Conversely, a quantum entanglement monotone of quantum measurements induces a quan…
Recently it has been shown that projected entangled-pair states can be considered as a (physically motivated) resource state for measurement-based quantum computation. Here we elaborate on how to construct a deterministic measurement-based quantum computation model on projected entangled-pair states. In particular, we use this scheme to build a 4-level many-body model for universal quantum computation on qubits. We also derive a 3-local parent Hamiltonian for this model, whose ground state is non-degenerate.
A special class of soft quantum measurements as a physical model of the fuzzy measurements widely used in physics is introduced and its information properties are studied in detail.
In this paper we investigate the linear and nonlinear models of optical quantum computation and discuss their scalability and efficiency. We show how there are significantly different scaling properties in single photon computation when weak cross-Kerr nonlinearities are allowed to supplement the usual linear optical set. In particular we show how quantum non-demolition measurements are an efficient resource for universal quantum computation.
The complementary roles played by parallel quantum computation and quantum measurement in originating the quantum speed-up are illustrated through an analogy with a famous metaphor by J.L. Borges.
In order to guarantee the output of a quantum computation, we usually assume that the component devices are trusted. However, when the total computation process is large, it is not easy to guarantee the whole system when we have scaling effects, unexpected noise, or unaccounted correlations between several subsystems. If we do not trust the measurement basis nor the prepared entangled state, we do need to be worried about such uncertainties. To this end, we proposes a "self-guaranteed" protocol for verification of quantum computation under the scheme of measurement-based quantum computation wh…
How important is fast measurement for fault-tolerant quantum computation? Using a combination of existing and new ideas, we argue that measurement times as long as even 1,000 gate times or more have a very minimal effect on the quantum accuracy threshold. This shows that slow measurement, which appears to be unavoidable in many implementations of quantum computing, poses no essential obstacle to scalability.
Vaidman, in a recent article adopts the method of 'quantum weak measurements in pre- and postselected ensembles' to ascertain whether or not the chained-Zeno counterfactual computation scheme proposed by Hosten et al. is counterfactual; which has been the topic of a debate on the definition of counterfactuality. We disagree with his conclusion, which brings up some interesting aspects of quantum weak measurements and some concerns about the way they are interpreted.
We develop a formal model for distributed measurement-based quantum computations, adopting an agent-based view, such that computations are described locally where possible. Because the network quantum state is in general entangled, we need to model it as a global structure, reminiscent of global memory in classical agent systems. Local quantum computations are described as measurement patterns. Since measurement-based quantum computation is inherently distributed, this allows us to extend naturally several concepts of the measurement calculus, a formal model for such computations. Our goal is …
We present a finite set of projective measurements that, together with quantum memory and preparation of the |0> state, suffice for universal quantum computation. This extends work of Nielsen [quant-ph/0108020], who proposed a scheme in which an arbitrary unitary operation on n qubits can be simulated using only projective measurements on at most 2n qubits. All measurements in our set involve two qubits, except two measurements which involve three qubits. Thus we improve by one the upper bound, implied by Nielsen's results, on the maximum number of qubits needed to participate in any single me…
What resources are universal for quantum computation? In the standard model, a quantum computer consists of a sequence of unitary gates acting coherently on the qubits making up the computer. This paper shows that a very different model involving only projective measurements, quantum memory, and the ability to prepare the |0> state is also universal for quantum computation. In particular, no coherent unitary dynamics are involved in the computation.
Measurement-based quantum computation with continuous variables in an optical setup shows the great promise towards implementation of large-scale quantum computation, where the time-domain multiplexing approach enables us to generate the large-scale cluster state used to perform measurement-based quantum computation. To make effective use of the advantage of the time-domain multiplexing approach, in this paper, we propose the method to generate the large-scale 3-dimensional cluster state which is a platform for topologically protected measurement-based quantum computation. Our method combines …
We show, under natural assumptions for qubit systems, that measurement-based quantum computations (MBQCs) which compute a non-linear Boolean function with high probability are contextual. The class of contextual MBQCs includes an example which is of practical interest and has a super-polynomial speedup over the best known classical algorithm, namely the quantum algorithm that solves the Discrete Log problem.
We show that the quantum dimer state on the Kagome lattice, which was recently realized with high fidelity in a Rydberg quantum simulator [Semeghini et al., Science 374, 6572 (2021)], offers a sufficient resource for universal measurement-based quantum computations. In particular, we provide an efficient encoding of logical qubits in this state, and give explicit measurement sequences that implement a universal set of gates on these qubits. Since the building blocks of the proposed measurements have already been experimentally implemented, our work highlights one possible path towards promotin…
In this paper, elementary quantum gate operations, such as the phase gate, the controlled-NOT gate, the swap and the Fredkin gate are constructed using joint measurement and pairs of entangled qubit pairs. The relation between the state of the entangled pair and the joint measurement basis is discussed. Some other generalization is also discussed.
Nielsen [quant-ph/0108020] introduced a model of quantum computation by measurement-based simulation of unitary computations. In this model, a consequence of the non-determinism of quantum measurement is the probabilistic termination of simulations. This means that the time when simulation terminates for a given computation is probabilistic, and this simulation may even never end. We introduce (section 3) a measurement-based model with non probabilistic termination, which permits, unlike existing models, to predict the time of termination. This new scheme is a modification of Nielsen's. After …
Measurement-based quantum computation is a novel model of quantum computing where universal quantum computation can be done with only local measurements on each particle of a quantum many-body state, which is called a resource state. One large difference of the measurement-based model from the circuit model is the existence of byproducts. In the circuit model, a desired unitary U can be implemented deterministically, whereas the measurement-based model implements BU, where B is an additional operator, which is called a byproduct. In order to compensate byproducts, following measurement angles …
We present a hybrid scheme for quantum computation that combines the modular structure of elementary building blocks used in the circuit model with the advantages of a measurement-based approach to quantum computation. We show how to construct optimal resource states of minimal size to implement elementary building blocks for encoded quantum computation in a measurement-based way, including states for error correction and encoded gates. The performance of the scheme is determined by the quality of the resource states, where within this error model we find a threshold of the order of 10% local …
Nielsen [quant-ph/0108020] showed that universal quantum computation is possible given quantum memory and the ability to perform projective measurements on up to 4-qubits. We describe an improved method that requires only 2-qubit measurements, which are both sufficient and necessary. We present a method to partially collapse the $C_k$-hierarchy in the indirect construction of unitary gates [Gottesman and Chuang, Nature, {\bf 402} 309 (1999)], and apply the method to find discrete universal sets of 2-qubit measurements.
We present unified, systematic derivations of schemes in the two known measurement-based models of quantum computation. The first model (introduced by Raussendorf and Briegel [Phys. Rev. Lett., 86, 5188 (2001)]) uses a fixed entangled state, adaptive measurements on single qubits, and feedforward of the measurement results. The second model (proposed by Nielsen [Phys. Lett. A, 308, 96 (2003)] and further simplified by Leung [Int. J. Quant. Inf., 2, 33 (2004)]) uses adaptive two-qubit measurements that can be applied to arbitrary pairs of qubits, and feedforward of the measurement results. The …
The ability to perform a universal set of quantum operations based solely on static resources and measurements presents us with a strikingly novel viewpoint for thinking about quantum computation and its powers. We consider the two major models for doing quantum computation by measurements that have hitherto appeared in the literature and show that they are conceptually closely related by demonstrating a systematic local mapping between them. This way we effectively unify the two models, showing that they make use of interchangeable primitives. With the tools developed for this mapping, we the…
Low depth measurement-based quantum computation with qudits ($d$-level systems) is investigated and a precise relationship between this powerful model and qudit quantum circuits is derived in terms of computational depth and size complexity. To facilitate this investigation a qudit `unbounded fan-out' circuit model, in which a qudit may be quantum-copied into an arbitrary number of ancillas in a single time-step, is introduced and shown to be capable of implementing interesting $n$-qudit unitaries in constant depth. A procedure for reducing the quantum computational depth in the measurement-ba…
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.
Testing measurement-based computational phases of quantum matter on a quantum processor
University of Chicago（芝加哥大学）；Pritzker School of Molecular Engineering, University of Chicago（普林茨克分子工程学院，芝加哥大学）；University of British Columbia（不列颠哥伦比亚大学）；Stewart Blusson Quantum Matter Institute, University of British Columbia（斯图尔特·布拉斯量子物质研究所，不列颠哥伦比亚大学）
We introduce an open-source software library Graphix, which optimizes and simulates measurement-based quantum computation (MBQC). By combining the measurement calculus with an efficient graph state simulator, Graphix allows the classical preprocessing of Pauli measurements in the measurement patterns, significantly reducing the number of operations required to perform the quantum computation while maintaining determinism. For a measurement pattern translated from a quantum circuit, this corresponds to the preprocessing of all Clifford gates, and this improvement in the one-way model is importa…
Quantum measurement is universal for quantum computation. The model of quantum computation introduced by Nielsen and further developed by Leung relies on a generalized form of teleportation. In order to simulate any n-qubit unitary transformation with this model, 4 auxiliary qubits are required. Moreover Leung exhibited a universal family of observables composed of 4 two-qubit measurements. We introduce a model of quantum computation via measurements only, relying on state transfer: state transfer only retains the part of teleportation which is necessary for computating. In order to simulate a…
A protocol of quantum dense coding protection of two qubits is proposed in amplitude damping (AD) channel using weak measurement and reversal measurement. It is found that the capacity of quantum dense coding under the weak measurement and reversal measurement is always greater than that without weak measurement and reversal measurement. When the protocol is applied, for the AD channels with different damping coefficient, the result reflects that quantum entanglement can be protected and quantum dense coding becomes successful.
We establish a connection between measurement-based quantum computation and the field of mathematical logic. We show that the computational power of an important class of quantum states called graph states, representing resources for measurement-based quantum computation, is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs. In particular, we show that for all graph state resources which can yield a computational speed-up with respect to classical computation, the underlying graphs--describing the quantum correlations of the s…
We propose a method to prepare entangled states and implement quantum computation with atoms in optical cavities. The internal state of the atoms are entangled by a measurement of the phase of light transmitted through the cavity. By repeated measurements an entangled state is created with certainty, and this entanglement can be used to implement gates on qubits which are stored in different internal degrees of freedom of the atoms. This method, based on measurement induced dynamics, has a higher fidelity than schemes making use of controlled unitary dynamics.
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.
Quantum measurements play a fundamental role in quantum mechanics and quantum information processing, but it is not easy to implement generalized measurements, the most powerful measurements allowed by quantum mechanics. Here we propose a simple recipe for implementing generalized measurements on a qudit via quantum walks. With this recipe, any discrete quantum measurement can be implemented via a one-dimensional discrete quantum walk; the number of steps is only two times the number of measurement outcomes. As an illustration, we present a unified solution for implementing arbitrary symmetric…
This paper shows that, in measurement-based quantum computation, it is possible to write any quantum oracle implementing a classical function in constant depth. The result is shown through the equivalence between MBQC and the circuit model where arbitrary rotations along $Z$ axis and unbounded fan-outs are elementary operations. A corollary of this result is that disjunction can be implemented exactly in constant-depth, answering an open question of Høyer and Špalek.
We study measurement-based quantum computation (MQC) using as quantum resource the planar code state on a two-dimensional square lattice (planar analogue of the toric code). It is shown that MQC with the planar code state can be efficiently simulated on a classical computer if at each step of MQC the sets of measured and unmeasured qubits correspond to connected subsets of the lattice.
Knowledge of optimal quantum measurements is important for a wide range of situations, including quantum communication and quantum metrology. Quantum measurements are usually optimised with an ideal experimental realisation in mind. Real devices and detectors are, however, imperfect. This has to be taken into account when optimising quantum measurements. In this paper, we derive the optimal minimum-cost and minimum-error measurements for a general model of imperfect detection.
One-way measurement based quantum computations (1WQC) may describe unitary transformations, via a composition of CPTP maps which are not all unitary themselves. This motivates the following decision problems: Is it possible to determine whether a ``quantum-to-quantum'' 1WQC procedure (having non-trivial input and output subsystems) performs a unitary transformation? Is it possible to describe precisely how such computations transform quantum states, by translation to a quantum circuit of comparable complexity? In this article, we present an efficient algorithm for transforming certain families…
Quantum computation offers a promising new kind of information processing, where the non-classical features of quantum mechanics can be harnessed and exploited. A number of models of quantum computation exist, including the now well-studied quantum circuit model. Although these models have been shown to be formally equivalent, their underlying elementary concepts and the requirements for their practical realization can differ significantly. The new paradigm of measurement-based quantum computation, where the processing of quantum information takes place by rounds of simple measurements on qubi…
We propose an enhanced discrimination measurement for tripartite 3-dimensional entangled states in order to improve the discernible number of orthogonal entangled states. The scheme suggests 3-dimensional Bell state measurement by exploiting composite two 3-dimensional state measurement setups. The setup relies on state-of-the-art techniques, a multi-port interferometer and nondestructive photon number measurements that are used for the post-selection of suitable ensembles. With this scheme, the sifted signal rate of measurement-device-independent quantum key distribution using 3-dimensional q…
Weak measurements may result in extra quantity of quantumness of correlations compared with standard projective measurement on a bipartite quantum state. We show that the quantumness of correlations by weak measurements can be consumed for information encoding which is only accessible by coherent quantum interactions. Then it can be considered as a resource for quantum information processing and can quantify this quantum advantage. We conclude that weak measurements can create more valuable quantum correlation.
An inequality about quantum noise is presented with the imprecise measurement theory, which is used to analyse the quantum limit in continuous quantum measurement. Different from the linear-response approach based on the quantum relation between noise and susceptibilities of the detector, we provide an explicit functional relation between quantum noise and reduction operator, and show a rigorous result: The minimum noise added by the detector in quantum measurement is precisely equal to the zero-point noise. This conclusion generalizes the standard Haus-Caves quantum limit for a linear amplifi…
Quantum mechanics, through the Heisenberg uncertainty principle, imposes limits to the precision of measurement. Conventional measurement techniques typically fail to reach these limits. Conventional bounds to the precision of measurements such as the shot noise limit or the standard quantum limit are not as fundamental as the Heisenberg limits, and can be beaten using quantum strategies that employ `quantum tricks' such as squeezing and entanglement.
A concept of the generalized quantum measurement is introduced as the transformation, which establishes a correspondence between the initial states of the object system and final states of the object--measuring device (meter) system with the help of a classical informational index, unambiguously linked to the classically compatible set of states of the object--meter system. It is shown that the generalized measurement covers all the key known quantum measurement concepts--standard projective, entangling, fuzzy and the generalized measurement with the partial or complete destruction of the init…
