D. Deutsch, A. Barenco, A. Ekert
We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates.
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Recently de La Torre et al. [1] reconstructed Quantum Theory from its local structure on the basis of local discriminability and the existence of a one-parameter group of bipartite transformations containing an entangling gate. This result relies on universality of an entangling gate for quantum computation. Here we prove universality of C-NOT with local gates for Real Quantum Theory (RQT), showing that such universality would not be sufficient for the result, whereas local discriminability and the qubit structure play a crucial role. For reversible computation, generally an extra rebit is nee…
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.
We analyze the critical quantum fluctuations in a coherently driven planar optical parametric oscillator. We show that the presence of transverse modes combined with quantum fluctuations changes the behavior of the `quantum image' critical point. This zero-temperature non-equilibrium quantum system has the same universality class as a finite-temperature magnetic Lifshitz transition.
Measurement-based quantum computing (MBQC) is a model of quantum computation where quantum information is coherently processed by means of projective measurements on highly entangled states. Following the introduction of MBQC, cluster states have been studied extensively both from the theoretical and experimental point of view. Indeed, the study of MBQC was catalysed by the realisation that cluster states are universal for MBQC with (X,Y)-plane and Z measurements. Here we examine the question of whether the requirement for Z measurements can be dropped while maintaining universality. We answer…
The common security criterion d in quantum key distribution is taken to solve the universal composability problem in quantum key distribution as well as providing good general quantitative security guarantee. In this paper it is shown that these are a result of an invalid interpretation of d. The general security significance of d is analyzed in detail. The related issues of universality and attacker's side information are discussed.
We study fidelity susceptibility in one-dimensional asymmetric Hubbard model, and show that the fidelity susceptibility can be used to identify the universality class of the quantum phase transitions in this model. The critical exponents are found to be 0 and 2 for cases of half-filling and away from half-filling respectively.
We build a framework allowing for a systematic investigation of the issue: "Which quantum states are universal resources for one-way quantum computation?" We start by re-examining what is exactly meant by "universality" in quantum computation, and what the implications are for universal one-way quantum computation. Given the framework of a measurement-based quantum computer, where quantum information is processed by local operations only, the most general universal one-way quantum computer is one which is capable of accepting arbitrary classical inputs and producing arbitrary quantum outputs--…
Dynamical formation of entanglement is studied for quantum chaotic bi-particle systems. We find that statistical properties of the Schmidt eigenvalues for strong chaos are well described by the random matrix theory of the Laguerre ensemble. This implies that entanglement formation for quantum chaos has universal properties, and does not depend on specific aspects of the systems.
We investigate for which resource states an efficient classical simulation of measurement based quantum computation is possible. We show that the Schmidt--rank width, a measure recently introduced to assess universality of resource states, plays a crucial role in also this context. We relate Schmidt--rank width to the optimal description of states in terms of tree tensor networks and show that an efficient classical simulation of measurement based quantum computation is possible for all states with logarithmically bounded Schmidt--rank width (with respect to the system size). For graph states …
We propose an effective set of elementary quantum gates which provide an encoded universality and demonstrate the physical feasibility of these gates for the solid-state quantum computer based on the multi-atomic systems in the QED cavity. We use the two-qubit encoding and swapping-based operations to simplify a physical realization of universal quantum computing and add the immunity to a number of errors. This approach allows to implement any encoded single-qubit operation by three elementary gates and the encoded controlled- NOT operation can be performed in a single step. The considerable a…
Unavoidable disturbance caused by a quantum measurement implies that the realizable subsequent measurements are getting limited after one performs some measurement. The obvious general limitation that one cannot circumvent by sequential or any other method is that the actually implemented measurements must be jointly measurable. In this work we show that any jointly measurable pair of observables can be obtained in a sequential measurement scheme, even if the second observable would be decided after the first measurement. This universality feature holds only for measurement schemes with a spec…
Recent work has demonstrated the existence of universal Hamiltonians - simple spin lattice models that can simulate any other quantum many body system to any desired level of accuracy. Until now proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians. In this work we go beyond this approach, and completely classify the simulation ability of quantum Hamiltonians by their complexity classes. We do this by deriving necessary and sufficient complexity theoretic conditions characterising universal quantum Hamiltonians. Although the re…
Universality in Terms of The BECs Bloch-Sphere Manipulations
本研究分析了BECs量子比特系统中的普适性，在SU(2)的((N+1))维表示下，考虑两种量子态表示，阐明了量子计算普适性的理论方面。
The quantum Fourier transform (QFT) is a powerful tool in quantum computing. The main ingredients of QFT are formed by the Walsh-Hadamard transform H and phase shifts P(.), both of which are 2x2 unitary matrices as operators on the two-dimensional 1-qubit space. In this paper, we show that H and P(.) suffice to generate the unitary group U(2) and, consequently, through controlled-U operations and their concatenations, the entire unitary group U(2^n) on n-qubits can be generated. Since any quantum computing algorithm in an n-qubit quantum computer is based on operations by matrices in U(2^n), i…
A Lie-algebraic Criterion for the Universality of Exponentiated Quantum Gates
本文提出一种多项式时间算法，用于判断由某些哈密顿量指数化的有限量子门集是否通用，基于李代数理论和Borel-de Siebenthal理论，揭示了量子门通用性与李代数表示不可约性的深刻联系。
We show that Coecke's compositionality theorem for quantum information flow follows by the universal property of tensor products from the case in which all relevant states are totally disentangled, for which the proof is almost trivial. With the same technique we deduce a PROP structure behind general multipartite quantum information processing and show that all such are equivalent to a canonical teleportation-type form. Some philosophical issues concerning quantum information are also touched upon.
Recently, a series of different measures quantifying memory effects in the quantum dynamics of open systems has been proposed. Here, we derive a mathematical representation for the non-Markovianity measure based on the exchange of information between the open system and its environment which substantially simplifies its numerical and experimental determination, and fully reveals the locality and universality of non-Markovianity in the quantum state space. We further illustrate the application of this representation by means of an all-optical experiment which allows the measurement of the degre…
Manin, Feynman, and Deutsch have viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum logic circuits and quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. The problem of universality has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani as well as Kitaev and Solovay. The quantum logic circuit model, developed by Feynman and Deutsch, has been more prominent in the research literature than Deutsch's quantum Turing mach…
We reason about possible future development of quantum game theory and its impact on information processing and the emerging information society. Two of the authors have recently proposed a quantum description of financial market in terms of quantum game theory. These "new games" cannot by themselves create extraordinary profits or multiplication of goods, but they may cause the dynamism of transaction which would result in more effective markets and capital flow into hands of the most efficient traders. We focus upon the problem of universality of measurement in quantum market games. Quantum-…
We revisit the question of universality in quantum computing and propose a new paradigm. Instead of forcing a physical system to enact a predetermined set of universal gates (e.g., single-qubit operations and CNOT), we focus on the intrinsic ability of a system to act as a universal quantum computer using only its naturally available interactions. A key element of this approach is the realization that the fungible nature of quantum information allows for universal manipulations using quantum information encoded in a subspace of the full system Hilbert space, as an alternative to using physical…
We consider a model of quantum computation we call "Varying-$Z$" (V$Z$), defined by applying controllable $Z$-diagonal Hamiltonians in the presence of a uniform and constant external $X$-field, and prove that it is universal, even in 1D. Universality is demonstrated by construction of a universal gate set with $O(1)$ depth overhead. We then use this construction to describe a circuit whose output distribution cannot be classically simulated unless the polynomial hierarchy collapses, with the goal of providing a low-resource method of demonstrating quantum supremacy. The V$Z$ model can achieve …
A theory of non-unitary-invertible as well as unitary canonical transformations is formulated in the context of Weyl's phase space representations. That all quantum canonical transformations without an explicit $\hbar$ dependence are also classical mechanical and vice versa is demonstrated in the phase space. Contrary to some earlier results, it is also shown that the quantum generators and their classical counterparts are identical and $\hbar$-independent. The latter is a powerful result bringing the theory of classical canonical transformations and the $\hbar$-independent quantum ones on an …
Given stabilizer operations and the ability to repeatedly prepare a single-qubit mixed state rho, can we do universal quantum computation? As motivation for this question, "magic state" distillation procedures can reduce the general fault-tolerance problem to that of performing fault-tolerant stabilizer circuits. We improve the procedures of Bravyi and Kitaev in the Hadamard "magic" direction of the Bloch sphere to achieve a sharp threshold between those rho allowing universal quantum computation, and those for which any calculation can be efficiently classically simulated. As a corollary, the…
It was shown by M.A. Nielsen and I.L. Chuang 1997, that it is impossible to build strictly universal programmable quantum gate array, that could perform any unitary operation precisely and it was suggested to use probabilistic gate arrays instead. In present work is shown, that if to use more physical and weak condition of universality (suggested already in earliest work by D.Deutsch 1985) and to talk about simulation with arbitrary, but finite precision, then it is possible to build universal programmable gate array. But now the same no-go theorem by Nielsen and Chuang will have new interesti…
On universality of hardware-efficient ansatzes
该研究针对硬件高效型变分电路（HEA），通过证明任何相关量子电路可高效表示为HEA电路，得出模拟HEA某些主要类别的问题是BQP完全的结论，揭示了HEA的普适性特征。
Quantum computation can be performed by encoding logical qubits into the states of two or more physical qubits, and controlling a single effective exchange interaction and possibly a global magnetic field. This "encoded universality" paradigm offers potential simplifications in quantum computer design since it does away with the need to perform single-qubit rotations. Here we show that encoded universality schemes can be combined with quantum error correction. In particular, we show explicitly how to perform fault-tolerant leakage correction, thus overcoming the main obstacle to fault-tolerant…
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 measuremen…
We analyze and compare the optimality of approximate and probabilistic universal programmable quantum processors. We define several characteristics how to quantify the optimality and we study in detail performance of three types of programmable quantum processors based on (1) the C-NOT gate, (2) the SWAP operation, and (3) the model of the quantum information distributor - the QID processor. We show under which conditions the measurement assisted QID processor is optimal. We also investigate optimality of the so-called U-processors and we also compare the optimal approximative implementation o…
Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This "magic states distillation" question is closely related to quantum fault tolerance. Lower bounds on the noise tolerable on the ancilla help give lower bounds on the tolerable noise rate threshold for fault-tolerant computation. Upper bounds show the limits of threshold upper-bound arguments based on the Gottesman-Knill theorem. We extend the range of single-qubit mixed states that are known to give univer…
The paper reviews and discusses four ideas scattered in previous papers of the author. First, objective properties of quantum systems are not associated with observables but are defined by preparations. Second, measurable results of classical theories are not sharp. All such results can be obtained as high-entropy limits of quantum mechanics. Third, a careful study of exchange symmetry requires a modification of the theory of measurement. Fourth, screens and detectors are closely associated with state reduction. A new, rather improved understanding of quantum mechanics is achieved without any …
We say that collection of $n$-qudit gates is universal if there exists $N_0\geq n$ such that for every $N\geq N_0$ every $N$-qudit unitary operation can be approximated with arbitrary precision by a circuit built from gates of the collection. Our main result is an upper bound on the smallest $N_0$ with the above property. The bound is roughly $d^8 n$, where $d$ is the number of levels of the base system (the '$d$' in the term qu$d$it.) The proof is based on a recent result on invariants of (finite) linear groups.
We show that the state-independent violation of inequalities for noncontextual hidden variable theories introduced in [Phys. Rev. Lett. 101, 210401 (2008)] is universal, i.e., occurs for any quantum mechanical system in which noncontextuality is meaningful. We describe a method to obtain state-independent violations for any system of dimension d > 2. This universality proves that, according to quantum mechanics, there are no "classical" states.
Measurement-based quantum computing is one of the most promising quantum computing models. Although various universal resource states have been proposed so far, it was open whether only two Pauli bases are enough for both of universal measurement-based quantum computing and its verification. In this paper, we construct a universal hypergraph state that only requires $X$ and $Z$-basis measurements for universal measurement-based quantum computing. We also show that universal measurement-based quantum computing on our hypergraph state can be verified in polynomial time using only $X$ and $Z$-bas…
Universality in Quantum Measurements
本文回顾了基于测量仪器环境诱导退相干的量子测量理论，总结出与幺正薛定谔演化一致的一般原理，并指出量子理论是对现象学实在的解释而非描述。
YZ-plane measurement-based quantum computation: Universality and Parity Architecture implementation
研究证明了在输入等于输出的统一确定性测量基量子计算中，必须使用YZ平面测量的图结构，并展示了YZ平面仅测量模式的通用性及与XZ平面模式的联系。
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, uni…
Universal quantum computation can be achieved by simply performing single-qubit measurements on a highly entangled resource state, such as cluster states. Cai, Miyake, Dür, and Briegel recently constructed a ground state of a two-dimensional quantum magnet by combining multiple Affleck-Kennedy-Lieb-Tasaki quasichains of mixed spin-3/2 and spin-1/2 entities and by mapping pairs of neighboring spin-1/2 particles to individual spin-3/2 particles [Phys. Rev. A 82, 052309 (2010)]. They showed that this state enables universal quantum computation by single-spin measurements. Here, we give an alterna…
We introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a counterexample to Ranchin's universality proof, and give another proof by Lie theory that the qudit ZX …
To generate arbitrary one- and two-qubit gates, the universal decompositions are usually used in quantum computing, and the universality of these decompositions has been demonstrated. However, in realistic experiments, gate errors may affect the universality of the universal decompositions. Here, we focus on the single-qubit-gate decomposition scheme and study the coherent-error effects on universality. We prove that, in the parameter space which we studied, some kinds of coherent errors will not affect the original universality, but some others will destroy it. We provide the definition and a…
Certain quantum codes allow logic operations to be performed on the encoded data, such that a multitude of errors introduced by faulty gates can be corrected. An important class of such operations are {\em transversal}, acting bitwise between corresponding qubits in each code block, thus allowing error propagation to be carefully limited. If any quantum operation could be implemented using a set of such gates, the set would be {\em universal}; codes with such a universal, transversal gate set have been widely desired for efficient fault-tolerant quantum computation. We study the structure of G…
Entropic Dynamics (ED) is a framework that allows the formulation of dynamical theories as an application of entropic methods of inference. In the generic application of ED to derive the Schroedinger equation for N particles the dynamics is a non-dissipative diffusion in which the system follows a "Brownian" trajectory with fluctuations superposed on a smooth drift. We show that there is a family of ED models that differ at the "microscopic" or sub-quantum level in that one can enhance or suppress the fluctuations relative to the drift. Nevertheless, members of this family belong to the same u…
Universality in the Anticoncentration of Chaotic Quantum Circuits
本文发现随机量子电路反集中现象可由一个仅含两个拟合参数的普适函数形式描述，该形式在有限尺寸和浅层深度下仍成立，并据此提出利用浅层电路对更大规模量子设备进行基准测试的实用框架。
Note on the Universality of Parameterized IQP Circuits with Hidden Units for Generating Probability Distributions
本文针对参数化IQP电路生成模型，给出两个简单证明，说明通过引入隐藏单元（被迹除的量子比特）可使其边缘分布具备生成任意概率分布的普适性，且所需额外量子比特数量合理。
Universality in fidelity-based quantum metrology
本文基于保真度准则提出几何方法，证明在量子计量中，存在与变换无关的零测度自旋态集合，其成员始终为最优或最差传感器。
We present a theoretical analysis of the paradigm of encoded universality, using a Lie algebraic analysis to derive specific conditions under which physical interactions can provide universality. We discuss the significance of the tensor product structure in the quantum circuit model and use this to define the conjoining of encoded qudits. The construction of encoded gates between conjoined qudits is discussed in detail. We illustrate the general procedures with several examples from exchange-only quantum computation. In particular, we extend our earlier results showing universality with the i…
Catalysing Completeness and Universality
该研究将量子催化应用于计算通用性证明与图形语言完备性扩展，简洁证明了CS+Hadamard、Toffoli+Hadamard门组的通用性，并扩展了无相位ZH-calculus的完备片段。
The Gottesman-Kitaev-Preskill (GKP) quantum error-correcting code has emerged as a key technique in achieving fault-tolerant quantum computation using photonic systems. Whereas [Baragiola et al., Phys. Rev. Lett. 123, 200502 (2019)] showed that experimentally tractable Gaussian operations combined with preparing a GKP codeword $\lvert 0\rangle$ suffice to implement universal quantum computation, this implementation scheme involves a distillation of a logical magic state $\lvert H\rangle$ of the GKP code, which inevitably imposes a trade-off between implementation cost and fidelity. In contrast…
Time evolution of impurity models and their universality for quantum computation
研究了杂质模型的时间演化在量子计算中的普遍性，证明了在特定条件下，时间演化能实现通用量子计算。
A general theory of preparational uncertainty relations for a quantum particle in one spatial dimension is developed. We derive conditions which determine whether a given smooth function of the particle's variances and its covariance is bounded from below. Whenever a global minimum exists, an uncertainty relation has been obtained. The squeezed number states of a harmonic oscillator are found to be universal: no other pure or mixed states will saturate any such relation. Geometrically, we identify a convex uncertainty region in the space of second moments which is bounded by the inequality der…
We propose a generalized Dicke model which supports a quantum tricritical point. We map out the phase diagram and investigate the critical behaviors of the model through exact low-energy effective Hamiltonian in the thermodynamic limit. As predicted by the Landau theory of phase transition, the order parameter shows non-universality at the tricritical point. Nevertheless, as a result of the separation of the classical and the quantum degrees of freedom, we find a universal relation between the excitation gap and the entanglement entropy for the entire critical line including the tricritical po…
Can quantum-mechanical particles propagating on a fixed spacetime background be approximated as test bodies satisfying the weak equivalence principle? We ultimately answer the question in the negative but find that, when universality of free-fall is assessed locally, a nontrivial agreement between quantum mechanics and the weak equivalence principle exists. Implications for mass sensing by quantum probes are discussed in some details.
We supply a rigorous proof that an open dense set of all possible 2-qubit gates G has the property that if the quantum circuit model is restricted to only permit swap of qubits lines and the application of G to pairs of lines, then the model is still computationally universal.
Dynamical quantum phase transitions (DQPTs) at critical times appear as non-analyticities during nonequilibrium quantum real-time evolution. Although there is evidence for a close relationship between DQPTs and equilibrium phase transitions, a major challenge is still to connect to fundamental concepts such as scaling and universality. In this work, renormalization group transformations in complex parameter space are formulated for quantum quenches in Ising models showing that the DQPTs are critical points associated with unstable fixed points of equilibrium Ising models. Therefore, these DQPT…
We study the Universality and Membership Problems for gate sets consisting of a finite number of quantum gates. Our approach relies on the techniques from compact Lie groups theory. We also introduce an auxiliary problem called Subgroup Universality Problem, which helps in solving some instances of the Membership Problem, and can be of interest on its own. The resulting theorems are mainly formulated in terms of centralizers and the adjoint representations of a given set of quantum gates.
Free-fall non-universality in quantum theory
本文将爱因斯坦等效原理与广义协变性纳入量子理论，证明波函数扩散导致量子粒子自由落体轨迹依赖其内部性质，并用厄特沃什参数给出可在原子干涉测量中检验的定量估计。
Bohr and von Neumann on the Universality of Quantum Mechanics: Materials for the History of the Quantum Measurement Process
针对学界对玻尔与冯·诺依曼量子测量观点解读争议大、误解多的问题，本文通过文本分析指出二者观点远较通常认知的接近，在量子力学普适性上立场一致，可助力完善量子测量问题史研究。
The paper is withdrawn by the author because it is superseded by cond-mat/0303357 .
Tuning the universality class of a quantum process by Trotterization
研究量子接触过程离散版本，通过数值分析发现其与连续时间版本相变所属普适类不同，差异源于离散化细节，构建插值方法，证明普适类修改源于相消干涉效应，揭示量子力学起源。
Using lie algebra, this brief text provides an upper bound on the universality of QAOA. That is, we prove that the upper bound for the number of alterations of QAOA required to approximate a universal gate set is within O(n)
We address the three points raised by the authors of the above Comment.
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