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In this paper we study universality for quantum gates acting on qudits.Qudits are states in a Hilbert space of dimension d where d is at least two. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit gates together with V produces all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions on V, and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor. We discuss some applications and also relations with work of other authors.

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... Single qubit gates do not suffice for universal quantum computation, and must be complemented with at least one entangling 2-qubit gate to achieve this universality [58]. Whenever we refer to universality in this thesis, we mean approximate universality where products of unitaries from a (usually finite) set of fixed unitaries can approximate any unitary up to arbitrary precision (see [58] for example for a more formal definition). ...

... Single qubit gates do not suffice for universal quantum computation, and must be complemented with at least one entangling 2-qubit gate to achieve this universality [58]. Whenever we refer to universality in this thesis, we mean approximate universality where products of unitaries from a (usually finite) set of fixed unitaries can approximate any unitary up to arbitrary precision (see [58] for example for a more formal definition). A particularly useful 2-qubit entangling gate which we will frequently use is the controlled Z gate, which is denoted as CZ and whose unitary matrix is given by ...

... Analytically, we show that almost any 4 assignment of fixed XY angle measurements on a n = 2 γ -row, 2-column cluster state (where γ is an integer) gives a random unitary set U B which is universal in U (2 n ). We use a Lie algebraic approach outlined by [58], and observations in [124,125] to prove this result. In particular, when γ = 1 we get that almost any assignment of fixed XY angle measurements generates universal sets U B ⊂ U (4), which in general can be invertible, partially invertible or non invertible. ...

This thesis is focused on the generation and understanding of particular kinds of quantum randomness. Randomness is useful for many tasks in physics and information processing, from randomized benchmarking , to black hole physics , as well demonstrating a so-called quantum speedup , and many other applications. On the one hand we explore how to generate a particular form of random evolution known as a t-design. On the other we show how this can also give instances for quantum speedup - where classical computers cannot simulate the randomness efficiently. We also show that this is still possible in noisy realistic settings. More specifically, this thesis is centered around three main topics. The first of these being the generation of epsilon-approximate unitary t-designs. In this direction, we first show that non-adaptive, fixed measurements on a graph state composed of poly(n,t,log(1/epsilon)) qubits, and with a regular structure (that of a brickwork state) effectively give rise to a random unitary ensemble which is a epsilon-approximate t-design. This work is presented in Chapter 3. Before this work, it was known that non-adaptive fixed XY measurements on a graph state give rise to unitary t-designs , however the graph states used there were of complicated structure and were therefore not natural candidates for measurement based quantum computing (MBQC), and the circuits to make them were complicated. The novelty in our work is showing that t-designs can be generated by fixed, non-adaptive measurements on graph states whose underlying graphs are regular 2D lattices. These graph states are universal resources for MBQC. Therefore, our result allows the natural integration of unitary t-designs, which provide a notion of quantum pseudorandomness which is very useful in quantum algorithms, into quantum algorithms running in MBQC. Moreover, in the circuit picture this construction for t-designs may be viewed as a constant depth quantum circuit, albeit with a polynomial number of ancillas. We then provide new constructions of epsilon-approximate unitary t-designs both in the circuit model and in MBQC which are based on a relaxation of technical requirements in previous constructions. These constructions are found in Chapters 4 and 5.

... Therefore, we can look for possible kinds of analogies in process of defining theoretic and practice conventions, rules and applications of the specific characteristics in elaboration quantum calculation strategies. We have not investigated possibilities to create directly quantum calculation units and practice calculation structures like qubits, registers, gates etc. [4,18], but dealing with spin and quantum definitions and descriptions we can try to involve these notices from different domains. Such a pragmatic approach only intuitively gives chances to create the transition theory and implement it even partially. ...

... We will start with a basic question: why to consider quantum computing? This involves introduction of basic ideas of quantum computing via a comparison of quantum and randomized computing [4,21]. Finite dimensional Hilbert spaces correspond to the example of such particle properties as spin value, polarization, position or momentum. ...

... Our proposition refers to a suggestion that outside magnetic field can play role of a controller of character logic operations, implemented by magnetic spin gates. Qubit is a quantum state of two level own orientations |0> and |1>, which can be presented as superposition of inner states: |Y> = a|0> + b|1> |a| 2 + |b| 2 = 1 (1) Qubit can be illustrated like spin in Bloch sphere ( Fig.1 and 2) [4]. Physically we can treat qubit as a spin in magnetic field being superposition of the base states. ...

Magnetic properties of spin glass materials [9,13] are close to quantum interpretation in their nature description [17]. Therefore, we can look for possible kinds of analogies in process of defining theoretic and practice conventions, rules and applications of the specific characteristics in elaboration quantum calculation strategies. We have not investigated possibilities to create directly quantum calculation units and practice calculation structures like qubits, registers, gates etc. [4,18], but dealing with spin and quantum definitions and descriptions we can try to involve these notices from different domains. Such a pragmatic approach only intuitively gives chances to create the transition theory and implement it even partially. Obviously, almost all of us have heard about quantum factorization, cryptography or teleportation but it is obtained as a result of exploration casually selected quantum properties and adapting them to mathematic problems. In our approach, we carefully investigate involutions among spin and quantum nature looking at possible implementation in molecular network.

... and the set of the Hadamard gate H and the T gate can generate all single-qubit gates. An entangling two-qubit gate [5,34] is defined as a two-qubit gate capable of transforming a tensor product of two single-qubit states into an entangling two-qubit state. For example, the CNOT gate is a maximally entangling gate [31], and it with single-qubit gates can generate the Bell states (3) from the product states. ...

... The set of an entangling two-qubit gate [34] with single-qubit gates is called a universal quantum gate set, with which universal quantum computation can be performed in the circuit model [1] of quantum computation. Hence the set of the CNOT gate (or the CZ gate) with singlequbit gates H and T forms a universal quantum gate set. ...

It is well-known that maximally entangled states such as the Greenberger-Horne-Zeilinger (GHZ) states, with the Bell states as the simplest examples, are widely exploited in quantum information and computation. We study the application of such maximally entangled states from the viewpoint of the GHZ transform, which is a unitary basis transformation from the product states to the GHZ states. The algebraic structure of the GHZ transform is made clear and representative examples for it are verified as multi-qubit Clifford gates. In this paper, we focus on the Bell transform as the simplest example of the GHZ transform and apply it to the reformulation of quantum circuit model of teleportation and the reformulation of the fault-tolerant construction of single-qubit gates and two-qubit gates in teleportation-based quantum computation. We clearly show that there exists a natural algebraic structure called the teleportation operator in terms of the Bell transform to catch essential points of quantum teleportation, and hence we expect that there would also exist interesting algebraic structures in terms of the GHZ transform to play important roles in quantum information and computation.

... An alternative construction of a universal quantum computer is derived from the entangling property of F s on 2-qudits. The work of Bremner et al.[12]and Brylinski and Brylinski[13]demonstrated that the inclusion of any 2-qudit entangling gate to the set of all single qudit gates is universal for quantum computation. Therefore we obtain the following universal gate set by including F s to the set of single qudit gates, {single qudit gates, F s }. ...

... An alternative construction of a universal quantum computer is derived from the entangling property of F s on 2-qudits. The work of Bremner et al. [11] and Brylinski and Brylinski [12] demonstrated that the inclusion of any 2-qudit entangling gate to the set of all single qudit gates is universal for quantum computation. Therefore we obtain the following universal gate set by including F s to the set of single qudit gates, {single qudit gates, F s }. (A.5) ...

We introduce diagrammatic protocols and holographic software for quantum information. We give a dictionary to translate between diagrammatic protocols and the usual algebraic protocols. In particular we describe the intuitive diagrammatic protocol for teleportation. We introduce the string Fourier transform $\mathfrak{F}_{s}$ in quantum information, which gives a topological quantum computer. We explain why the string Fourier transform maps the zero particle state to the multiple-qudit resource state, which maximizes the entanglement entropy. We give a protocol to construct this $n$-qudit resource state $|Max \rangle$, which uses minimal cost. We study Pauli $X,Y,Z$ matrices, and their relation with diagrammatic protocols. This work provides bridges between the new theory of planar para algebras and quantum information, especially in questions involving communication in quantum networks.

... Another challenge for qudit-based quantum processors is that algorithms designed for qubit-based systems must be modified and adapted to fully exploit the increased Hilbert space provided by qudits. The algorithms must be compiled using a qudit gate set, which differs from the qubit gate set [29][30][31][32]. Some well-known quantum algorithms, such as the Deutsch-Jozsa algorithm [33,34], Bernstein-Vazirani algorithm [35], Grover's algorithm [36], Quantum Fourier Transform [37,38], and Shor's algorithm [39], have direct generalizations of the qubit counterparts, which maintain the same principles but change the positional notation from the base-2 numeral system to the base-d numeral system. ...

Using quantum systems with more than two levels, or qudits, can scale the computation space of quantum processors more efficiently than using qubits, which may offer an easier physical implementation for larger Hilbert spaces. However, individual qudits may exhibit larger noise, and algorithms designed for qubits require to be recompiled to qudit algorithms for execution. In this work, we implemented a two-qubit emulator using a 4-level superconducting transmon qudit for variational quantum algorithm applications and analyzed its noise model. The major source of error for the variational algorithm was readout misclassification error and amplitude damping. To improve the accuracy of the results, we applied error-mitigation techniques to reduce the effects of the misclassification and qudit decay event. The final predicted energy value is within the range of chemical accuracy. Our work demonstrates that qudits are a practical alternative to qubits for variational algorithms.

... With a larger state space per subsystem, qudits offer potential advantages for quantum communication [6], quantum algorithms [7-10], and topological quantum systems [11][12][13] . Quantum computation with qudits can also reduce circuit complexity and can be advantageous in a variety of NISQ-era applications [8-10, 14,15]. Qudits may also provide significantly advantages in quantum error correction and fault-tolerant quantum computation [16][17][18][19][20]. ...

We study the generation of two-qudit entangling quantum logic gates using two techniques in quantum optimal control. We take advantage of both continuous, Lie-algebraic control and digital, Lie-group control. In both cases, the key is access to a time-dependent Hamiltonian which can generate an arbitrary unitary matrix in the group SU(d 2). We find efficient protocols for creating high-fidelity entangling gates. As a test of our theory, we study the case of qudits robustly encoded in nuclear spins of alkaline earth atoms and manipulated with magnetic and optical fields, with entangling interactions arising from the well-known Rydberg blockade. We applied this in a case study based on a d = 10 dimensional qudit encoded in the I = 9/2 nuclear spin in 87 Sr, controlled through a combination of nuclear spin-resonance, a tensor AC-Stark shift, and Rydberg dressing, which allows us to generate an arbitrary symmetric entangling two-qudit gate such as CPhase. Our techniques can be used to implement qudit entangling gates for any 2 ≤ d ≤ 10 encoded in the nuclear spin. We also studied how decoherence due to the finite lifetime of the Rydberg states affects the creation of the CPhase gate and found, through numerical optimization, a fidelity of 0.9985, 0.9980, 0.9942, and 0.9800 for d = 2, d = 3, d = 5, and d = 7 respectively. This provides a powerful platform to explore the various applications of quantum information processing of qudits including metrological enhancement with qudits, quantum simulation, universal quantum computation, and quantum error correction.

... Quantum gates exploit the quantum entanglement and superposition states of qubits [24,25]. In quantum computing, the state of a qubit is changed with a quantum gate that can perform reversible operations. ...

In this paper, we propose a quantum circuit for the SPEEDY block cipher for the first time and estimate its security strength based on the post-quantum security strength presented by NIST. The strength of post-quantum security for symmetric key cryptography is estimated at the cost of the Grover key retrieval algorithm. Grover’s algorithm in quantum computers reduces the n-bit security of block ciphers to n2 bits. The implementation of a quantum circuit is required to estimate the Grover’s algorithm cost for the target cipher. We estimate the quantum resource required for Grover’s algorithm by implementing a quantum circuit for SPEEDY in an optimized way and show that SPEEDY provides either 128-bit security (i.e., NIST security level 1) or 192-bit security (i.e., NIST security level 3) depending on the number of rounds. Based on our estimated cost, increasing the number of rounds is insufficient to satisfy the security against quantum attacks on quantum computers.

... To study this problem we use a Lie-algebraic approach, which has been previously used to study universality in the absence of symmetries [2,15,16,[33][34][35][36][37]. Suppose one can implement Hamiltonians j a j (t)A j , where {a j } are arbitrary real functions of time and {A j } are Hermitian operators. ...

Universality of local unitary transformations is one of the cornerstones of quantum computing with many applications and implications that go beyond this field. However, it has been recently shown that this universality does not hold in the presence of continuous symmetries: generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems [I. Marvian, Nature Physics (2022)]. In this work, we study qubit circuits formed from k-local rotationally-invariant unitaries and fully characterize the constraints imposed by locality on the realizable unitaries. We also present an interpretation of these constraints in terms of the average energy of states with a fixed angular momentum. Interestingly, despite these constraints, we show that, using a pair of ancilla qubits, any rotationally-invariant unitary can be realized with the Heisenberg exchange interaction, which is 2-local and rotationally-invariant. We also show that a single ancilla is not enough to achieve universality. Finally, we discuss applications of these results for quantum computing with semiconductor quantum dots, quantum reference frames, and resource theories.

... Toutes portes non réversible ne pourront donc pasêtre simulé par l'évolution d'un qubit. A la différence des bits classiques, il existe une famille de transformations (unitaire ou binaire) qui permettent de réaliser des opérations logiques sur les qubits [40,66] : ...

La principale contribution de cette thèse est d'étudier expérimentalement le comportement des ordinateurs quantiques analogiques tels que ceux commercialisés par D-Wave lorsqu'ils sont confrontés à des cas de problèmes de couplage biparti de cardinalité maximale spécifiquement conçus pour être difficiles à résoudre au moyen d'un recuit simulé. Nous comparons un "Washington" (2X) de D-Wave avec 1098 qubits utilisables sur différentes tailles d'instances et nous observons que pour tous ces cas, sauf les plus triviaux, la machine ne parvient pas à obtenir une solution optimale. Ainsi, nos résultats suggèrent que le recuit quantique, du moins tel qu'il est mis en œuvre dans un dispositif D-Wave, tombe dans les mêmes pièges que le recuit simulé et fournit donc des preuves supplémentaires suggérant qu'il existe des problèmes polynomiaux qu'une telle machine ne peut pas résoudre efficacement pour atteindre l'optimalité. En outre, nous étudions dans quelle mesure les topologies d'interconnexion des qubits expliquent ces derniers résultats expérimentaux.

... Toutes portes non réversible ne pourront donc pasêtre simulé par l'évolution d'un qubit. A la différence des bits classiques, il existe une famille de transformations (unitaire ou binaire) qui permettent de réaliser des opérations logiques sur les qubits [40,66] : ...

The main contribution of this thesis is to investigate the behavior of analog quantum computers as commercialized by D-Wave when confronted to instances of the maximum cardinality matching problem which is specifically designed to be hard to solve by means of simulated annealing. We benchmark a D-Wave “Washington” (2X) with 1098 operational qubits on various sizes of such instances and observe that for all but the most trivially small of these it fails to obtain an optimal solution. Thus, our results suggest that quantum annealing, at least as implemented in a D-Wave device, falls in the same pitfalls as simulated annealing and hence provides additional evidences suggesting that there exist polynomial-time problems that such a machine cannot solve efficiently to optimality. Additionally, we investigate the extent to which the qubits interconnection topologies explains these latter experimental results.

... In this work, we extend Grover's algorithm to any dimension, hence comes the concept of qudit. Qudit technology is that quantum technology, which deals with d-ary quantum system, where d ≥ 2 [3]. For providing larger state space, simultaneous multiple control operations, we graduate to qudits which eventually reduce the circuit complexity and aggrandize the efficiency of quantum algorithms [4,5,6]. ...

As the development of Quantum computers becomes reality, the implementation of quantum algorithms is accelerating in a great pace. Grover's algorithm in a binary quantum system is one such quantum algorithm which solves search problems with numeric speed-ups than the conventional classical computers. Further, Grover's algorithm is extended to a $d$-ary quantum system for utilizing the advantage of larger state space. In qudit or $d$-ary quantum system n-qudit Toffoli gate plays a significant role in the accurate implementation of Grover's algorithm. In this paper, a generalized $n$-qudit Toffoli gate has been realized using qudits to attain a logarithmic depth decomposition without ancilla qudit. Further, the circuit for Grover's algorithm has been designed for any d-ary quantum system, where d >= 2, with the proposed $n$-qudit Toffoli gate so as to get optimized depth as compared to state-of-the-art approaches. This technique for decomposing an n-qudit Toffoli gate requires access to higher energy levels, making the design susceptible to leakage error. Therefore, the performance of this decomposition for the unitary and erasure models of leakage noise has been studied as well.

... The unitary operators can act as quantum gates, however not all may lead to a universal set of gates. According to a theorem by Brylinski [30] for a 2-qubit space, a gate helps building a universal set if and only if it is entangling. We can use this criterion to check which of the operators in Table 1 are entangling and can potentially lead to a universal set. ...

Unitary braiding operators can be used as robust entangling quantum gates. We introduce a solution-generating technique to solve the ( d , m , l ) -generalized Yang-Baxter equation, for m / 2 ≤ l ≤ m , which allows to systematically construct such braiding operators. This is achieved by using partition algebras, a generalization of the Temperley-Lieb algebra encountered in statistical mechanics. We obtain families of unitary and non-unitary braiding operators that generate the full braid group. Explicit examples are given for a 2-, 3-, and 4-qubit system, including the classification of the entangled states generated by these operators based on Stochastic Local Operations and Classical Communication.

... A set of universal quantum gates is a set of quantum gates that can, in a finite sequence of gates from this set, replicate any arbitrary unitary operation that may be possible on a quantum computer [70,[281][282][283]69]. For physical systems with exchange interaction, universal quantum gates have been constructed with encoded qubits [25,284], while the Loss-Divincenzo Quantum Computer relies on the √ SWAP and single-qubit gates [115]. ...

This project is a comprehensive investigation into the application of the exchange interaction, particularly with the realization of the SWAP^1/n quantum operator, in quantum information processing. We study the generation, characterization and application of entanglement in such systems. Given the non-commutativity of neighbouring SWAP^1/n gates, the mathematical study of combinations of these gates is an interesting avenue of research that we have explored, though due to the exponential scaling of the complexity of the problem with the number of qubits in the system, numerical techniques, though good for few-qubit systems, are found to be inefficient for this research problem when we look at systems with higher number of qubits. Since the group of SWAP^1/n operators is found to be isomorphic to the symmetric group Sn, we employ group-theoretic methods to find the relevant invariant subspaces and associated vector-states. Some interesting patterns of states are found including onedimensional invariant subspaces spanned by W-states and the Hamming-weight preserving symmetry of the vectors spanning the various invariant subspaces. We also devise new ways of characterizing entanglement and approach the separability problem by looking at permutation symmetries of subsystems of quantum states. This idea is found to form a bridge with the entanglement characterization tool of Peres-Horodecki’s Partial Positive Transpose (PPT), for mixed quantum states. We also look at quantum information taskoriented ‘distance’ measures of entanglement, besides devising a new entanglement witness in the ‘engle’. In terms of applications, we define five different formalisms for quantum computing: the circuit-based model, the encoded qubit model, the cluster-state model, functional quantum computation and the qudit-based model. Later in the thesis, we explore the idea of quantum computing based on decoherence-free subspaces. We also investigate ways of applying the SWAP^1/n in entanglement swapping for quantum repeaters, quantum communication protocols and quantum memory.

... The unitary operators can act as quantum gates, however not all may lead to a universal set of gates. According to a theorem by Brylinski [30] for a 2-qubit space, a gate helps building a universal set if and only if it is entangling. We can use this criterion to check which of the operators in Table 1 are entangling and can potentially lead to a universal set. ...

Unitary braiding operators can be used as robust entangling quantum gates. We introduce a solution-generating technique to solve the $(d,m,l)$-generalized Yang-Baxter equation, for $m/2\leq l \leq m$, which allows to systematically construct such braiding operators. This is achieved by using partition algebras, a generalization of the Temperley-Lieb algebra encountered in statistical mechanics. We obtain families of unitary and non-unitary braiding operators that generate the full braid group. Explicit examples are given for a 2-, 3-, and 4-qubit system, including the classification of the entangled states generated by these operators based on Stochastic Local Operations and Classical Communication.

... A set of universal quantum gates is a set of quantum gates that can, in a finite sequence of gates from this set, replicate any arbitrary unitary operation that may be possible on a quantum computer [49] [50] [51] [52]. For physical systems with exchange interaction, universal quantum gates have been constructed with encoded qubits [26] [53], while the Loss-Divincenzo Quantum ...

In this paper, I propose new models of quantum information processing using the exchange interaction in physical systems. The partial SWAP operator that can be realized using the exchange interaction is used as the underlying resource for defining models of quantum computation, quantum communication, quantum memory and decoherence-free subspaces. Given the non-commutativity of these operators (for adjacent operators operating on a common qubit), a number of quantum states and entanglement patters can be obtained. This zoo of states can be classified, due to the parity constraints and permutation symmetry of the states, into invariant subspaces that are used for the definition of some of the applications in this paper.

... In what follows Hilbert spaces are always finite-dimensional. In virtue of the Choi isomorphism, this theorem is equivalent to the fact that the only channels on a system of two qudits that preserve the set of maximally entangled states are products of local unitaries and swap [59,60]. ...

In general relativity, the causal structure between events is dynamical, but it is definite and observer-independent; events are point-like and the membership of an event A in the future or past light-cone of an event B is an observer-independent statement. When events are defined with respect to quantum systems however, nothing guarantees that the causal relationship between A and B is definite. We propose to associate a causal reference frame corresponding to each event, which can be interpreted as an observer-dependent time according to which an observer describes the evolution of quantum systems. In the causal reference frame of one event, this particular event is always localised, but other events can be 'smeared out' in the future and in the past. We do not impose a predefined causal order between the events, but only require that descriptions from different reference frames obey a global consistency condition. We show that our new formalism is equivalent to the pure process matrix formalism (Araújo et al 2017 Quantum 1 10). The latter is known to predict certain multipartite correlations, which are incompatible with the assumption of a causal ordering of the events - these correlations violate causal inequalities. We show how the causal reference frame description can be used to gain insight into the question of realisability of such strongly non-causal processes in laboratory experiments. As another application, we use causal reference frames to revisit a thought experiment Zych et al (arXiv:1708.00248) where the gravitational time dilation due to a massive object in a quantum superposition of positions leads to a superposition of the causal ordering of two events. © 2018 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft.

... In virtue of the Choi isomorphism, this theorem is equivalent to the fact that the only channels on a system of two qudits that preserve the set of maximally entangled states are products of local unitaries and swap [42,43]. ...

In general relativity, the causal structure between events is dynamical, but it is definite and observer-independent; events are point-like and the membership of an event A in the future or past light-cone of an event B is an observer-independent statement. When events are defined with respect to quantum systems however, nothing guarantees that event A is definitely in the future or in the past of B. We propose to associate a causal reference frame corresponding to each event, which can be interpreted as an observer-dependent time according to which an observer describes the evolution of quantum systems. In the causal reference frame of one event, this particular event is always localised, but other events can be "smeared out" in the future and in the past. We do not impose a predefined causal order between the events, but only require that descriptions from different reference frames obey a global consistency condition. We show that our new formalism is equivalent to the pure process matrix formalism. The latter is known to predict certain multipartite correlations, which are incompatible with the assumption of a causal ordering of the events -- these correlations violate causal inequalities. We show how the causal reference frame description can be used to gain insight into the question of realisability of such strongly non-causal processes in laboratory experiments. As another application, we use causal reference frames to revisit a thought experiment where the gravitational time dilation due to a massive object in a quantum superposition of positions leads to a superposition of the causal ordering of two events.

... In [5], the Brylinskis give a general criterion of G to be universal. They prove that a two-qubit gate G is universal if and only if it is entangling. ...

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations as- sociated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.

... This section focuses on the last step in our approach, that is to show the qudit cluster-like state is universal for MBQC. Brylinski and Brylinski [47] proved the theorem that to simulate all possible gates, one only needs to simulate all possible one-qudit gates and one imprimitive two-qudit gate. A two-qudit gate is defined as primitive if it maps any decomposable state to another decomposable state. ...

Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically non-trivial, short-ranged entanged states are promising candidates for such a resource. Miller and Miyake [NPJ Quantum Information 2, 16036 (2016)] recently constructed a particular $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-protected topological state on the Union-Jack lattice and established its quantum computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting $(d-1)$ qudit nontrivial $\mathbb{Z}_d \times \mathbb{Z}_d \times \mathbb{Z}_d$ symmetry-protected topological states are universal on the triangular lattice, for $d$ being a prime number greater than 2. The same construction also holds for other 3-colorable lattices, including the Union-Jack lattice.

... In [5], the Brylinskis give a general criterion of G to be universal. They prove that a two-qubit gate G is universal if and only if it is entangling. ...

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.

... To prove that postselected CCC's can perform universal quantum computation, we need to show how to perform an entangling two qubit gate. We can then appeal to the result of Brylinski & Brylinski [BB02] and Bremner et al. [BDD + 02] that any entangling two-qubit gate, plus the set of all-one-qubit gates, is universal for quantum computation. But performing entangling two-qubit gates is trivial in our setup, since the Clifford group (and the conjugated Clifford group) contains entangling two-qubit gates. ...

A well-known result of Gottesman and Knill states that Clifford circuits - i.e. circuits composed of only CNOT, Hadamard, and $\pi/4$ phase gates - are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) - where one additionally conjugates every qubit by the same one-qubit gate U - can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be simulated classically to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, we show that this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture.

... From our earlier discussion, the only states in H GS with the same (minimal) entanglement entropy as the basis states are those described by a factorizable Ψ α1,...,α N D . The form ofĝ| GS is therefore strongly constrained, and in general we can write [35]ĝ ...

The Lieb-Schultz-Mattis (LSM) theorem and its extensions forbid trivial phases from arising in certain quantum magnets. Constraining infrared behavior with the ultraviolet data encoded in the microscopic lattice of spins, these theorems are particularly important because they tie the absence of spontaneous symmetry breaking to the emergence of exotic phases like quantum spin liquids. In this work, we take a new topological perspective on these theorems, by arguing they originate from an obstruction to "trivializing" the lattice under smooth, symmetric deformations. We refer to the study of such deformations as the "lattice homotopy problem." We further conjecture that all LSM-like theorems for quantum magnets (many previously-unknown) can be understood from lattice homotopy, which automatically incorporates the full spatial symmetry group of the lattice, including all its point-group symmetries. To substantiate the claim, we prove the conjecture in two dimensions for some physically relevant settings.

... Alternatively, a controlled-NOT gate could be employed. This is because in combination with single-qubit gates it forms a set of universal gates with which any unitary operation can be performed [332]. In fact, a controlled-NOT gate has already been implemented using on-demand single photons [288]. ...

The development of integrated quantum photonics is integral to many areas of quantum information science, in particular linear optical quantum computing. In this context, a diversity of physical systems is being explored and thus versatility and adaptability are important prerequisites for any candidate platform. Silicon oxynitride is a promising material because its refractive index can be varied over a wide range. This dissertation describes the development of silicon oxynitride waveguides for applications in the field of integrated quantum photonics. The project consisted of three stages: design, characterisation, and application. First, the parameter space was studied through simulations. The structures were optimised to achieve low-loss devices with a small footprint at a wavelength of 900 nm. Buried channel waveguides with a cross-section of 1.6 μm x 1.6 μm and a core (cladding) refractive index of 1.545 (1.505) were chosen. Second, following their fabrication with plasma-enhanced chemical vapour deposition, electron beam lithography, and reactive ion etching, the waveguides were characterised. The refractive index was shown to be tunable from the silica to the silicon nitride regime. Optimised tapers significantly improved the coupling efficiency. The minimum bend radius was measured to be less than 2 mm. Propagation losses as low as 1.45 dB cm-1 were achieved. Directional couplers with coupling ratios ranging from 0 to 1 were realised. Third, building blocks for linear optical quantum computing were demonstrated. Reconfigurable quantum circuits consisting of Mach-Zehnder interferometers with near perfect visibilities were fabricated along with a four-port switch. The potential of quantum speedup was illustrated by carrying out the Deutsch-Jozsa algorithm with a fidelity of 100 % using on-demand single photons from a quantum dot. This dissertation presents the first implementation of tunable Mach-Zehnder interferometers, which act on single photons, based on silicon oxynitride waveguides. Furthermore, for the first time silicon oxynitride photonic quantum circuits were operated with on-demand single photons. Accordingly, this work has created a platform for the development of integrated quantum photonics.

Arising from a special session held at the 2010 North American Annual Meeting of the Association for Symbolic Logic, this volume is an international cross-disciplinary collaboration with contributions from leading experts exploring connections across their respective fields. Themes range from philosophical examination of the foundations of physics and quantum logic, to exploitations of the methods and structures of operator theory, category theory, and knot theory in an effort to gain insight into the fundamental questions in quantum theory and logic. The book will appeal to researchers and students working in related fields, including logicians, mathematicians, computer scientists, and physicists. A brief introduction provides essential background on quantum mechanics and category theory, which, together with a thematic selection of articles, may also serve as the basic material for a graduate course or seminar.

Quantum circuit complexity has played a central role in recent advances in holography and many‐body physics. Within quantum field theory, it has typically been studied in a Lorentzian (real‐time) framework. In a departure from standard treatments, we aim to quantify the complexity of the Euclidean path integral. In this setting, there is no clear separation between space and time, and the notion of unitary evolution on a fixed Hilbert space no longer applies. As a proof of concept, we argue that the pants decomposition provides a natural notion of circuit complexity within the category of 2‐dimensional bordisms and use it to formulate the circuit complexity of states and operators in 2‐dimensional topological quantum field theory. We comment on analogies between our formalism and others in quantum mechanics, such as tensor networks and second quantization.

In this article, we demonstrate that, by employing the OpenPulse design kit for IBM superconducting quantum devices, the controlled-V gate (
cv
gate) can be implemented in about half the gate time to the controlled-X gate (
cx
or
cnot
gate) and consequently 65.5% reduced gate time compared to the
cx
-based implementation of
cv
. Then, based on the theory of Cartan decomposition, we characterize the set of all two-qubit gates implemented with only two or three
cv
gates; using pulse-engineered
cv
gates, enables us to implement these gates with shorter gate time and possibly better gate fidelity than the
cx
-based one, as actually demonstrated in two examples. Moreover, we showcase the improvement of linearly coupled three-qubit Toffoli gate by implementing it with the pulse-engineered
cv
gate, both in gate time and the averaged output-state fidelity. These results imply the importance of our
cv
gate implementation technique, which, as an additional option for the basis gate set design, may shorten the overall computation time and consequently improve the precision of several quantum algorithms executed on a real device.

This chapter provides a brief description of quantum machine learning (QML) and its correlation with artificial intelligence. It shows how the quantum counterpart of machine learning (ML) is much faster and more efficient than classical ML. In the QML techniques, the chapter develops quantum algorithms to operate the classical algorithms on a quantum computer. The quantum decision tree employs quantum states to create the classifiers used in ML. The current generation of quantum computing technologies calls for quantum algorithms that require a limited number of qubits and quantum gates, and that are robust against errors. The chapter discusses a low‐depth variational quantum algorithm for supervised learning. The parameter‐shift rule is an approach to evaluating gradients of parameterized quantum circuits on quantum hardware. The chapter introduces a quantum neural network that can represent labeled data – classical or quantum – and be trained by supervised learning.

According to a fundamental result in quantum computing, any unitary transformation on a composite system can be generated using so-called 2-local unitaries that act only on two subsystems. Beyond its importance in quantum computing, this result can also be regarded as a statement about the dynamics of systems with local Hamiltonians: although locality puts various constraints on the short-term dynamics, it does not restrict the possible unitary evolutions that a composite system with a general local Hamiltonian can experience after a sufficiently long time. Here we show that this universality does not remain valid in the presence of conservation laws and global continuous symmetries such as U(1) and SU(2). In particular, we show that generic symmetric unitaries cannot be implemented, even approximately, using local symmetric unitaries. Based on this no-go theorem, we propose a method for experimentally probing the locality of interactions in nature. In the context of quantum thermodynamics, our results mean that generic energy-conserving unitary transformations on a composite system cannot be realized solely by combining local energy-conserving unitaries on the components. We show how this can be circumvented via catalysis.

There is currently a large interest in understanding the potential advantages quantum devices can offer for probabilistic modelling. In this work we investigate, within two different oracle models, the probably approximately correct (PAC) learnability of quantum circuit Born machines, i.e., the output distributions of local quantum circuits. We first show a negative result, namely, that the output distributions of super-logarithmic depth Clifford circuits are not sample-efficiently learnable in the statistical query model, i.e., when given query access to empirical expectation values of bounded functions over the sample space. This immediately implies the hardness, for both quantum and classical algorithms, of learning from statistical queries the output distributions of local quantum circuits using any gate set which includes the Clifford group. As many practical generative modelling algorithms use statistical queries -- including those for training quantum circuit Born machines -- our result is broadly applicable and strongly limits the possibility of a meaningful quantum advantage for learning the output distributions of local quantum circuits. As a positive result, we show that in a more powerful oracle model, namely when directly given access to samples, the output distributions of local Clifford circuits are computationally efficiently PAC learnable by a classical learner. Our results are equally applicable to the problems of learning an algorithm for generating samples from the target distribution (generative modelling) and learning an algorithm for evaluating its probabilities (density modelling). They provide the first rigorous insights into the learnability of output distributions of local quantum circuits from the probabilistic modelling perspective.

Under the actions of different Hamiltonians on the different two-qubit input states by using the quantum Yang-Baxterization approach, we investigate the behaviors of the fidelity and the trace distance as measures of ‘closeness’ and distinguishability of two quantum states. The results show that the fidelity that is the main figure of merit for any communication and computing process can be kept to high values depending on the choice of the initial states and the Hamiltonians constructed by the Yang-Baxter equation. On the other hand, by choosing the initial states and Yang-Baxter systems which are the various extensions of the Yang-Baxter equations for several matrices, these quantifiers can be adjusted as desired to achieve many quantum computing and computational tasks. Furthermore, to quantify the performance of quantum teleportation we examine the teleportation fidelity for the outputs that correspond to the different two-qubit X-type states under the actions of the different Hamiltonians. It is possible to obtain high fidelity to use the quantum teleportation process.

In this thesis we expand upon the results that led to the paper of Lee et al., arXiv:2105.01114 (2021). In particular, we give more details on the oracular formulation of variational quantum algorithms, and the relationship between properties of Ans\"atze and the strength of their corresponding oracles. Furthermore, having identified the importance of noncommutativity in parameterized quantum circuits (PQCs) as likely being crucial to achieving a quantum advantage, we compare this notion to similar properties in classical neural networks such as nonlinearity, based on the perspective of the recent moniker for PQCs as quantum neural networks. While this thesis includes much of the figures and content from the aforementioned paper, it should be considered mainly as a self-contained collection of supplementary materials.

The objects of study of this thesis are the origins of the quantum computational speed- up. For the past three decades research in quantum foundations pointed to a few dif- ferent properties of quantum systems that could be linked to computational power. We start our study investigating the power of correlations, as it is intrinsically found in the measurement-based model of quantum computation. An important recent contribu- tion to the field showed that measurements on three-qubit GHZ states lead to universal classical computation. In that scenario, a client initially limited to compute only sums modulo-2 can deterministically evaluate a non-linear (NAND) function when control- ling measurements on a GHZ state. We were interested in achieving deterministic computation of maximally non-linear functions using the same type of resource. Another interesting result related to the computation of a NAND function using GHZ states shows that it is possible to achieve the same task with unitary transfor- mations performed on a single qubit. Differently than in the protocol that uses GHZ states, in the single-qubit one, non-locality and traditional forms of contextuality can- not be linked to the computational advantage. In this thesis, we address the question of which type of non-classicality gives us the same computational power in the single- qubit scheme. We analyse carefully chosen variations of the protocol in terms of Bell’s and Tsirelson’s bounds and detect a connection between reversibility in transformations and the computational capability of the system.

Qudit is a multi-level computational unit alternative to the conventional 2-level qubit. Compared to qubit, qudit provides a larger state space to store and process information, and thus can provide reduction of the circuit complexity, simplification of the experimental setup and enhancement of the algorithm efficiency. This review provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building, algorithm design, to experimental methods. We first discuss the qudit gate universality and a variety of qudit gates including the pi/8 gate, the SWAP gate, and the multi-level controlled-gate. We then present the qudit version of several representative quantum algorithms including the Deutsch-Jozsa algorithm, the quantum Fourier transform, and the phase estimation algorithm. Finally we discuss various physical realizations for qudit computation such as the photonic platform, iron trap, and nuclear magnetic resonance.

Building a quantum computer that surpasses the computational power of its classical counterpart is a great engineering challenge. Quantum software optimizations can provide an accelerated pathway to the first generation of quantum computing applications that might save years of engineering effort. Current quantum software stacks follow a layered approach similar to the stack of classical computers, which was designed to manage the complexity. In this review, we point out that greater efficiency of quantum computing systems can be achieved by breaking the abstractions between these layers. We review several works along this line, including two hardware-aware compilation optimizations that break the quantum Instruction Set Architecture (ISA) abstraction and two error-correction/information-processing schemes that break the qubit abstraction. Last, we discuss several possible future directions.

The quantum approximate optimization algorithm (QAOA) is considered to be one of the most promising approaches towards using near-term quantum computers for practical application. In its original form, the algorithm applies two different Hamiltonians, called the mixer and the cost Hamiltonian, in alternation with the goal being to approach the ground state of the cost Hamiltonian. Recently, it has been suggested that one might use such a set-up as a parametric quantum circuit with possibly some other goal than reaching ground states. From this perspective, a recent work (Lloyd, arXiv:1812.11075) argued that for one-dimensional local cost Hamiltonians, composed of nearest neighbour ZZ terms, this set-up is quantum computationally universal and provides a universal gate set, i.e. all unitaries can be reached up to arbitrary precision. In the present paper, we complement this work by giving a complete proof and the precise conditions under which such a one-dimensional QAOA might produce a universal gate set. We further generalize this type of gate-set universality for certain cost Hamiltonians with ZZ and ZZZ terms arranged according to the adjacency structure of certain graphs and hypergraphs.

The notion of universal quantum computation can be generalized to multilevel qudits, which offer advantages in resource usage and algorithmic efficiencies. Trapped ions, which are pristine and well-controlled quantum systems, offer an ideal platform to develop qudit-based quantum information processing. Previous work has not fully explored the practicality of implementing trapped-ion qudits accounting for known experimental error sources. Here, we describe a universal set of protocols for state preparation, single-qudit gates, a generalization of the Mølmer-Sørensen gate for two-qudit gates, and a measurement scheme which utilizes shelving to a metastable state. We numerically simulate known sources of error from previous trapped-ion experiments, and show that there are no fundamental limitations to achieving fidelities above 99% for three-level qudits encoded in Ba+137 ions. Our methods are extensible to higher-dimensional qudits, and our measurement and single-qudit gate protocols can achieve 99% fidelities for five-level qudits. We identify avenues to further decrease errors in future work. Our results suggest that three-level trapped-ion qudits will be a useful technology for quantum information processing.

Since Yang-Baxter systems are easy to prepare and manipulate in quantum information experiments they are of increasing interest in the estimation of physical parameters. We investigate the dynamics of quantum Fisher information for the optimal estimation of parameters using two-qubit pure and different mixed states under action of the Yang-Baxter matrices. Although quantum Fisher information is monotonically decreasing under the action of a quantum channel, we have shown that mitigation of these decreases providing relative enhancements in quantum Fisher information is possible by means of Yang-Baxter matrices which model uniter quantum channels or noises.

Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qutrits. Past work with qutrits has demonstrated only constant factor improvements, owing to the log2(3) binary-to-ternary compression factor. We present a novel technique using qutrits to achieve a logarithmic depth (runtime) decomposition of the Generalized Toffoli gate using no ancilla-a significant improvement over linear depth for the best qubit-only equivalent. Our circuit construction also features a 70x improvement in two-qudit gate count over the qubit-only equivalent decomposition. This results in circuit cost reductions for important algorithms like quantum neurons and Grover search. We develop an open-source circuit simulator for qutrits, along with realistic near-term noise models which account for the cost of operating qutrits. Simulation results for these noise models indicate over 90% mean reliability (fidelity) for our circuit construction, versus under 30% for the qubit-only baseline. These results suggest that qutrits offer a promising path towards scaling quantum computation.

This paper is an introduction to relationships between topology, quantum computing, and the properties of Fermions. In particular, we study the remarkable unitary braid group representations associated with Majorana fermions. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

We construct a quantum reference frame, which can be used to approximately implement arbitrary unitary transformations on a system in the presence of any number of extensive conserved quantities, by absorbing any back action provided by the conservation laws. Thus, the reference frame at the same time acts as a battery for the conserved quantities. Our construction features a physically intuitive, clear and implementation-friendly realization. Indeed, the reference system is composed of the same types of subsystems as the original system and is finite for any desired accuracy. In addition, the interaction with the reference frame can be broken down into two-body terms coupling the system to one of the reference frame subsystems at a time. We apply this construction to quantum thermodynamic set-ups with multiple, possibly non-commuting conserved quantities, which allows for the definition of explicit batteries in such cases.
This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to arXiv:1708.01645, which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions $(d_1, d_2, \dots, d_n)$, and computes the dimension of the space ${\cal H}_{LME}/K$ of LME states up to local unitary transformations for all non-empty cases. In this paper, we provide a pedagogical overview and physical interpretation of the the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions $(2,A,B)$ and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results also give the dimension of the space of SLOCC equivalence classes for states with "generic" entanglement for all multipart systems since this space is equivalent to ${\cal H}_{LME}/K$. Finally, we give the dimension of the stabilizer subgroup $S \subset SL(d_1, \mathbb{C}) \times \cdots \times SL(d_n, \mathbb{C})$ for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

Although qubits are the leading candidate for the basic elements in a quantum computer, there are also a range of reasons to consider using higher-dimensional qudits or quantum continuous variables (QCVs). In this paper, we use a general “quantum variable” formalism to propose a method of quantum computation in which ancillas are used to mediate gates on a well-isolated “quantum memory” register and which may be applied to the setting of qubits, qudits (for d>2), or QCVs. More specifically, we present a model in which universal quantum computation may be implemented on a register using only repeated applications of a single fixed two-body ancilla-register interaction gate, ancillas prepared in a single state, and local measurements of these ancillas. In order to maintain determinism in the computation, adaptive measurements via a classical feed forward of measurement outcomes are used, with the method similar to that in measurement-based quantum computation (MBQC). We show that our model has the same hybrid quantum-classical processing advantages as MBQC, including the power to implement any Clifford circuit in essentially one layer of quantum computation. In some physical settings, high-quality measurements of the ancillas may be highly challenging or not possible, and hence we also present a globally unitary model which replaces the need for measurements of the ancillas with the requirement for ancillas to be prepared in states from a fixed orthonormal basis. Finally, we discuss settings in which these models may be of practical interest.

Recent work has shown that quantum computers can compute scattering probabilities in massive quantum field theories, with a run time that is polynomial in the number of particles, their energy, and the desired precision. Here we study a closely related quantum field-theoretical problem: estimating the vacuum-to-vacuum transition amplitude, in the presence of spacetime-dependent classical sources, for a massive scalar field theory in (1+1) dimensions. We show that this problem is BQP-hard; in other words, its solution enables one to solve any problem that is solvable in polynomial time by a quantum computer. Hence, the vacuum-to-vacuum amplitude cannot be accurately estimated by any efficient classical algorithm, even if the field theory is very weakly coupled, unless BQP=BPP. Furthermore, the corresponding decision problem can be solved by a quantum computer in a time scaling polynomially with the number of bits needed to specify the classical source fields, and this problem is therefore BQP-complete. Our construction can be regarded as an idealized architecture for a universal quantum computer in a laboratory system described by massive phi^4 theory coupled to classical spacetime-dependent sources.

In this paper, we investigate the ground-state fidelity and fidelity susceptibility in the many-body Yang–Baxter system and analyze their connections with quantum phase transition. The Yang–Baxter system was perturbed by a twist of eⁱφ at each bond, where the parameter φ originates from the q-deformation of the braiding operator U with q= e⁻ⁱφ (Jimbo in Yang–Baxter equations in integrable systems, World Scientific, Singapore, 1990), and φ has a physical significance of magnetic flux (Badurek et al. in Phys. Rev. D 14:1177, 1976). We test the ground-state fidelity related by a small parameter variation φ which is a different term from the one used for driving the system toward a quantum phase transition. It shows that ground-state fidelity develops a sharp drop at the transition. The drop gets sharper as system size N increases. It has been verified that a sufficiently small value of φ used has no effect on the location of the critical point, but affects the value of F(gc, φ). The smaller the twist φ, the more the value of F(gc, φ) is close to 0. In order to avoid the effect of the finite value of φ, we also calculate the fidelity susceptibility. Our results demonstrate that in the Yang–Baxter system, the quantum phase transition can be well characterized by the ground-state fidelity and fidelity susceptibility in a special way.

While the role of ternary reversible and quantum computation has been growing, synthesis methodologies for such logic, have been addressed in only a few works. A reversible ternary logic function can be expressed as minterms by using projection operators. In this paper, a novel realization of the projection operators using a minimum number of permutative ternary Muthukrishnan-Stroud (M-S) gates is presented. Next, an efficient method for logic simplification for ternary reversible logic is proposed. This method along with the new construction of projection operators yields significantly lower gate cost of approximately 31% less than that obtained by earlier methodologies, for the synthesis of ternary benchmark circuits.

This paper is an introduction to relationships between topology, quantum computing and the properties of fermions. In particular we study the remarkable unitary braid group representations associated with Majorana fermions.

In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structure of collectivies of electrons, as in the quantum Hall effect, but also in the structure of single electrons both by experiments with electrons in nanowires and also by the decomposition of the operator algebra for a fermion into a Clifford algebra generated by two Majorana operators. The purpose of this paper is to discuss these braiding representations, important for relationships among physics, quantum information and topology. A new result in this paper is the Clifford Braiding Theorem. This theorem shows that the Majorana operators give rise to a particularly robust representation of the braid group that is then further represented to find the phases of the fermions under their exchanges in a plane space. The more robust representation in our braiding theorem will be the subject of further work.

In the quest to build a practical quantum computer, it is important to use efficient schemes for enacting the elementary quantum operations from which quantum computer programs are constructed. The opposing requirements of well-protected quantum data and fast quantum operations must be balanced to maintain the integrity of the quantum information throughout the computation. One important approach to quantum operations is to use an extra quantum system – an ancilla – to interact with the quantum data register. Ancillas can mediate interactions between separated quantum registers, and by using fresh ancillas for each quantum operation, data integrity can be preserved for longer. This review provides an overview of the basic concepts of the gate model quantum computer architecture, including the different possible forms of information encodings – from base two up to continuous variables – and a more detailed description of how the main types of ancilla-mediated quantum operations provide efficient quantum gates.

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.

We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values $(x,y)$ to $(x,x \oplus y)$) is universal in the sense that all unitary operations on arbitrarily many bits $n$ (U($2^n$)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates, the asymptotic number required for $n$-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary $n$-bit unitary operations. Comment: 31 pages, plain latex, no separate figures, submitted to Phys. Rev. A. Related information on http://vesta.physics.ucla.edu:7777/

The theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines. Quantum gates are the generalization of classical logic gates. A single type of gate, the univeral quantum gate, together with quantum 'unit wires', is adequate for constructing networks with any possible quantum computational property.

It is shown that if one can apply some Hamiltonian repeatedly to a few variables at a time one can in general effect any desired unitary time evolution on an arbitrarily large number of variables. As a result, almost any quantum logic gate with two or more inputs is computationally universal in that copies of the gate can be ``wired together'' to effect any desired logic circuit, and to perform any desired unitary transformation on a set of quantum variables.

A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of three-bit gates, by analogy to the universality of the Toffoli three-bit gate of classical reversible computing. Two-bit quantum gates may be implemented by magnetic resonance operations applied to a pair of electronic or nuclear spins. A "gearbox quantum computer" proposed here, based on the principles of atomic force microscopy, would permit the operation of such two-bit gates in a physical system with very long phase breaking (i.e., quantum phase coherence) times. Simpler versions of the gearbox computer could be used to do experiments on Einstein-Podolsky-Rosen states and related entangled quantum states. 1994 PACS: 03.65.Bz, 89.80.+h, 02.20.Sv, 76.70.Fz Typeset using REVT E X I. INTRODUCTION Eventually computational devices will stop...

We prove the existence of a class of two--input, two--output gates any one of
which is universal for quantum computation. This is done by explicitly
constructing the three--bit gate introduced by Deutsch [Proc.~R.~Soc.~London.~A
{\bf 425}, 73 (1989)] as a network consisting of replicas of a single two--bit
gate.

Almost any quantum logic gate is universal University Park 16802 E-mail address: jlb@math.psu.edu and rkb@math.psu.edu URL: www

- S Lloyd

S. Lloyd, Almost any quantum logic gate is universal, Phys. Rev. Letters 75 (1995), 346-349 Department of Mathematics, Penn State University, University Park 16802 E-mail address: jlb@math.psu.edu and rkb@math.psu.edu URL: www.math.psu.edu/rkb 10