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.
An original presentation of Categorical Quantum Physics, in the line of Abramsky and Coecke, tries to introduce only objects and assumptions that are clearly relevant to Physics and does not assume compact closure. Adjoint arrows, tensor products and biproducts are the ingredients of this presentation. Tensor products are defined, up to a unitary arrow, by a universal property related to transformations of composite systems, not by assuming a monoidal structure. Entangled states of a tensor product define mixed states on the components of the tensor product. Coproducts that fit the adjoint str…
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 …
University of Oxford（牛津大学）；Queen Mary University of London（伦敦玛丽女王大学）
In both quantum mechanics and corpus linguistics based on vector spaces, the notion of entanglement provides a means for the various subsystems to communicate with each other. In this paper we examine a number of implementations of the categorical framework of Coecke, Sadrzadeh and Clark (2010) for natural language, from an entanglement perspective. Specifically, our goal is to better understand in what way the level of entanglement of the relational tensors (or the lack of it) affects the compositional structures in practical situations. Our findings reveal that a number of proposals for verb…
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…
Holographic Bit Threads from String-Diagrammatic Quantum Information Flow
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）
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 de…
Random-Matrix-Induced Simplicity Bias in Over-parameterized Variational Quantum Circuits
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA（电子与计算机工程系，佐治亚理工学院）；NVIDIA Research, Santa Clara, CA 95051, USA（NVIDIA研究）；IBM Thomas J. Watson Research Center, NY, 10598, USA（IBM托马斯·J·沃森研究中心）；Hon Hai (Foxconn) Quantum Computing Research Center, Taipei, 114, Taiwan（鸿海（富士康）量子计算研究中心）
Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern …
We show that gauge invariant quantum link models, Abelian and non-Abelian, can be exactly described in terms of tensor networks states. Quantum link models represent an ideal bridge between high-energy to cold atom physics, as they can be used in cold-atoms in optical lattices to study lattice gauge theories. In this framework, we characterize the phase diagram of a (1+1)-d quantum link version of the Schwinger model in an external classical background electric field: the quantum phase transition from a charge and parity ordered phase with non-zero electric flux to a disordered one with a net …
We provide an efficient approximation for the exponential of a local operator in quantum spin systems using tensor-network representations of a cluster expansion. We benchmark this cluster tensor network operator (cluster TNO) for one-dimensional systems, and show that the approximation works well for large real- or imaginary-time steps. We use this formalism for representing the thermal density operator of a two-dimensional quantum spin system at a certain temperature as a single cluster TNO, which we can then contract by standard contraction methods for two-dimensional tensor networks. We ap…
The variational tensor network renormalization approach to two-dimensional (2D) quantum systems at finite temperature is applied for the first time to a model suffering the notorious quantum Monte Carlo sign problem --- the orbital $e_g$ model with spatially highly anisotropic orbital interactions. Coarse-graining of the tensor network along the inverse temperature $β$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension $D$ --- limiting the amount of entanglement --- is a natural refinement parameter. Increasing $D$ we obtain a converged order para…
QuanTA: Efficient High-Rank Fine-Tuning of LLMs with Quantum-Informed Tensor Adaptation
NSF AI Institute for Artificial Intelligence and Fundamental Interactions（NSF人工智能与基础交互AI研究所）；Massachusetts Institute of Technology（麻省理工学院）；Harvard University（哈佛大学）
The most important feature of two-dimensional quantum Rokhsar-Kivelson (RK) type models is that their ground state wavefunction norms can be mapped into the partition functions of two-dimensional statistical models so that the quantum phase transitions become the thermal phase transitions of the corresponding statistical models. For a generic quantum critical point, we generalize the framework of RK wavefunctions by introducing the concept of the thermofield double (TFD) state, which is a purification of the equilibrium density operator. Moreover, by expressing the TFD state in terms of the pr…
The previously developed algebraic lightfront holography is used in conjunction with the tensor splitting of the chiral theory on the causal horizon. In this way a universal area law for the entanglement entropy of the vacuum relative to the split (tensor factorized) vacuum is obtained. The universality of the area law is a result of the kinematical structure of the properly defined lightfront degrees of freedom. We consider this entropy associated with causal horizon of the wedge algebra in Minkowski spacetime as an analog of the quantum Bekenstein black hole entropy similar to the way in whi…
