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One-way quantum computing achieves the full power of quantum computation by
performing single particle measurements on some many-body entangled state,
known as the resource state. As single particle measurements are relatively
easy to implement, the preparation of the resource state becomes a crucial
task. An appealing approach is simply to cool a...
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Citations
... Since the cluster state of one-way quantum computing does not occur naturally in the ground state of a physical system, the effort is to find an alternative resource state that appears as the ground state of spin lattices. Their results indicate that one-way quantum computing is not possible with a naturally occurring qubit system which is sufficient enough to ask whether there exists a naturally occurring state as a universal resource in the physical system[13]. ...
I attempt to investigate Hilary Putnam's view on the logic of realism as quantum logic. And what is inductive in such a logic that doesn't need any empirical advances from Boolean logical connectives. This means that, if quantum is true logic, then the state associated with a true logic is complete and also possesses the completeness of Boolean logic. Thus, investigating whether the completeness exists prior to the experimenter's choice i,e., an act of measurement, then one could also deduce a mapping from the complete state of a system to a monoidal category which is the system's classical counterpart. I call the mapping a naturally occurring state in a non-trivial group of Boolean logic. Thereby studying the logic of the experimental propositions and the system's classical counterparts through a reformulated realism based categories. Interestingly, if this study fails to fit with Putnam's original view, whenever a logician deduces propositions that arouse with meaningful statements of the experimenter's choice, any act of measurement might also means the completeness of quantum theory.
... We denote n(σ, τ ) with [ n] u = n u (σ, τ ) as the syndrome vector corresponding to σ⊗τ . If for different combination of (σ, τ ), their syndrome vectors are all distinct, then the states σ ⊗ τ |C are orthonormal and the cluster state |C cannot be an eigenstate of the Hamiltonian H [36,64]. Chen et al. used a different approach and gave a general proof for such a no-go theorem for any qubit twobody frustration-free Hamiltonian [65]. The proof uses the equivalence of quantum states under stochastic local operation and classical communication and the result by Bravyi on the homogeneous Hamiltonians [66]. ...
Measurement-based quantum computation is different from other approaches for quantum computation, in that everything needs to be done is only local measurement on a certain entangled state. It thus uses entanglement as the resource that drives computation. We give a pedagogical treatment on the basics, and then review some selected developments beyond graph states, including Affleck-Kennedy-Lieb-Tasaki states and more recent 2D symmetry-protected topological states.
... Importantly, graph states [118] or, more generally, stabilizer states such as the cluster state [119] are included in this class. Graph states can be approximated as ground states of two-body Hamiltonians [120], although it has been shown for spin-1/2 that this approximation cannot be made exact (ground states of frustration-free 2-local qubit Hamiltonians are unentangled [121,122]), even if we drop the frustration-freeness condition [123,124]. On the other hand, frustration-free, noncommuting models include the Affleck-Kennedy-Lieb-Tasaki (AKLT) [125], Rokhsar-Kivelson models [126,127] and parent Hamiltonians that are defined from injective projected entangledpair states (PEPS) [73,[128][129][130][131]. ...
Computationally intractable tasks are often encountered in physics and optimization. Such tasks often comprise a cost function to be optimized over a so-called feasible set, which is specified by a set of constraints. This may yield, in general, to difficult and non-convex optimization tasks. A number of standard methods are used to tackle such problems: variational approaches focus on parameterizing a subclass of solutions within the feasible set; in contrast, relaxation techniques have been proposed to approximate it from outside, thus complementing the variational approach by providing ultimate bounds to the global optimal solution. In this work, we propose a novel approach combining the power of relaxation techniques with deep reinforcement learning in order to find the best possible bounds within a limited computational budget. We illustrate the viability of the method in the context of finding the ground state energy of many-body quantum systems, a paradigmatic problem in quantum physics. We benchmark our approach against other classical optimization algorithms such as breadth-first search or Monte-Carlo, and we characterize the effect of transfer learning. We find the latter may be indicative of phase transitions, with a completely autonomous approach. Finally, we provide tools to generalize the approach to other common applications in the field of quantum information processing.
... For qubits (d = 2), frustration-free Local Hamiltonians always have a ground state in the form of a product state of one qubit and two qubit states [11,47,18]. For qutrits (d = 3), the AKLT model with a 2-dimensional MPS description of the ground state is well known. ...
... The rules of the PF model (18) allow us to change the color of an adjacent letter pair, e.g. . . . 11 · · · ↔ . . . ...
Investigating translationally invariant qudit spin chains with a low local dimension, we ask what is the best possible tradeoff between the scaling of the entanglement entropy of a large block and the inverse-polynomial scaling of the spectral gap. Restricting ourselves to Hamiltonians with a "rewriting" interaction, we find the pair-flip model, a family of spin chains with nearest neighbor, translationally invariant, frustration-free interactions, with a very entangled ground state and an inverse-polynomial spectral gap. For a ground state in a particular invariant subspace, the entanglement entropy across a middle cut scales as $\log n$ for qubits (it is equivalent to the XXX model), while for qutrits and higher, it scales as $\sqrt{n}$. Moreover, we conjecture that this particular ground state can be made unique by adding a small translationally-invariant perturbation that favors neighboring letter pairs, adding a small amount of frustration, while retaining the entropy scaling.
... Much is known about local FF quantum spin−1/2 chains. For example, the ground state entanglement entropy is zero 8 and their energy gap has been classified. 6 In general less is known for higher spin models. ...
We present exact results on the exactly solvable spin chain of Bravyi et al. [Phys. Rev. Lett. 109, 207202 (2012)]. This model is a spin one chain and has a Hamiltonian that is local and translationally invariant in the bulk. It has a unique (frustration free) ground state with an energy gap that is polynomially small in the system’s size (2n). The half-chain entanglemententropy of the ground state is 12logn+const.[Bravyi et al., Phys. Rev. Lett. 109, 207202 (2012)]. Here we first write the Hamiltonian in the standard spin-basis representation. We prove that at zero temperature, the magnetization is along the z-direction, i.e., ⟨sx⟩=⟨sy⟩=0 (everywhere on the chain). We then analytically calculate ⟨sz⟩ and the two-point correlation functions of sz. By analytically diagonalizing the reduced density matrices, we calculate the Schmidt rank, von Neumann, and Rényi entanglemententropies for the following: 1. Any partition of the chain into two pieces (not necessarily in the middle) and 2. L consecutive spins centered in the middle. Further, we identify entanglement Hamiltonians (Eqs. (49) and (59)). We prove a small lemma (Lemma (1)) on the combinatorics of lattice paths using the reflection principle to relate and calculate the Motzkin walk “height” to spin expected values. We also calculate the, closely related (scaled), correlation functions of Brownian excursions. The known features of this model are summarized in a table in Sec. I.
... We show that this is indeed the case, at least for a restricted set of clauses. Chen et al. [14] showed that for every YES instance of Quantum 2-SAT, there is always a satisfying assignment that is a product of single-and two-qubit states. In fact, with the restricted clause set that we consider, there will be a satisfying single-qubit product state of the form: ...
We study a quantum algorithm that consists of a simple quantum Markov process, and we analyze its behavior on restricted versions of Quantum 2-SAT. We prove that the algorithm solves this decision problem with high probability for n qubits, L clauses, and promise gap c in time O(n^2 L^2 c^{-2}). If the Hamiltonian is additionally polynomially gapped, our algorithm efficiently produces a state that has high overlap with the satisfying subspace. The Markov process we study is a quantum analogue of Sch\"oning's probabilistic algorithm for k-SAT.
... The equalities are results of Fenner et al. [23] and Aaronson [1], and the containments EQP ⊆ LWPP and BQP ⊆ AWPP are proven by Fortnow and Rogers [24]. and 2-QUANTUM-SAT [14,16,10] (in contrast to 2-LOCAL-HAMILTONIAN [25]). However, the study of EQP seems to be hampered by the restriction to finite gate-sets, which makes it difficult to produce exact algorithms for natural problems, except for some in the oracle model (e.g. ...
We present characterisations of "exact" gap-definable classes, in terms of
indeterministic models of computation which slightly modify the standard model
of quantum computation. This follows on work of Aaronson
[arXiv:quant-ph/0412187], who shows that the counting class PP can be
characterised in terms of bounded-error "quantum" algorithms which use
invertible (and possibly non-unitary) transformations, or postselections on
events of non-zero probability. Our work considers similar modifications of the
quantum computational model, but in the setting of exact algorithms, and
algorithms with zero error and constant success probability. We show that the
gap-definable counting classes [J. Comput. Syst. Sci. 48 (1994), p.116] which
bound exact and zero-error quantum algorithms can be characterised in terms of
"quantum-like" algorithms involving nonunitary gates, and that postselection
and nonunitarity have equivalent power for exact quantum computation only if
these classes collapse.
... One might hope for a two-body Hamiltonian with a graph state as a unique ground state, however, for spin-1/2 Hamiltonians, this has been proven impossible [23,24]. Frustration-free Hamiltonians are even more restrictive: the ground space of any frustration-free twobody spin-1/2 Hamiltonian is unentangled [25,26]. ...
... We first note that it is not possible to obtain an exact qubit graph state as a unique ground state of a two-body frustration-free Hamiltonian. This is known for qubit Hamiltonians [26], and we can easily extend this result to two-body frustration-free Hamiltonians on higher dimensional particles in the case where each graph state qubit resides in a two-dimensional subspace (which we will call the logical subspace) of each physical particle. The reasoning for this is as follows. ...
The framework of measurement-based quantum computation (MBQC) allows us to
view the ground states of local Hamiltonians as potential resources for
universal quantum computation. A central goal in this field is to find models
with ground states that are universal for MBQC and that are also natural in the
sense that they involve only two-body interactions and have a small local
Hilbert space dimension. Graph states are the original resource states for
MBQC, and while it is not possible to obtain graph states as exact ground
states of two-body Hamiltonians here we construct two-body frustration-free
Hamiltonians that have arbitrarily good approximations of graph states as
unique ground states. The construction involves taking a two-body
frustration-free model that has a ground state convertible to a graph state
with stochastic local operations, then deforming the model such that its ground
state is close to a graph state. Each graph state qubit resides in a subspace
of a higher dimensional particle. This deformation can be applied to two-body
frustration-free Affleck-Kennedy-Lieb-Tasaki (AKLT) models, yielding
Hamiltonians that are exactly solvable with exact tensor network expressions
for ground states. For the star-lattice AKLT model, the ground state of which
is not expected to be a universal resource for MBQC, applying such a
deformation appears to enhance the computational power of the ground state,
promoting it to a universal resource for MBQC. Transitions in computational
power, similar to percolation phase transitions, can be observed when
Hamiltonians are deformed in this way. Improving the fidelity of the ground
state comes at the cost of a shrinking gap. While analytically proving gap
properties for these types of models is difficult in general, we provide a
detailed analysis of the deformation of a spin-1 AKLT state to a linear graph
state.
... Then, the above equation can be interpreted that the ''gate operation'' [23][24][25][26] .) In particular, there is severe lack of knowledge about FT quantum computation on resource states with d $ 3. It is necessary to consider resource states with d $ 3 if we want to enjoy the cooling preparation of a resource state and the energy-gap protection of measurement-based quantum computation with a physically natural Hamiltonian, since no genuinely entangled qubit state can be the unique ground state of a two-body frustration-free Hamiltonian 27 . ...
In the framework of quantum computational tensor network, which is a general framework of measurement-based quantum computation, the resource many-body state is represented in a tensor-network form (or a matrix-product form), and universal quantum computation is performed in a virtual linear space, which is called a correlation space, where tensors live. Since any unitary operation, state preparation, and the projection measurement in the computational basis can be simulated in a correlation space, it is natural to expect that fault-tolerant quantum circuits can also be simulated in a correlation space. However, we point out that not all physical errors on physical qudits appear as linear completely-positive trace-preserving errors in a correlation space. Since the theories of fault-tolerant quantum circuits known so far assume such noises, this means that the simulation of fault-tolerant quantum circuits in a correlation space is not so straightforward for general resource states.
... Unfortunately, it was recently shown by Chen et al. that this may not be possible. Chen et al. provided a no-go theorem showing that there does not exist a unique ground state for a two-body frustration-free Hamiltonian that can simultaneously be a resource state for MBQC [32] for a qubit system. It was further shown by Ji et al. that indeed for two-level systems , the structure of the ground-state space for any two-body frustration-free Hamiltonian can be fully characterized [33] and none of these states corresponds to a resource state for MBQC. ...
... As the most practical quantum systems, it is natural to ask whether it is possible to find ideal resource states for MBQC from qubit systems. Unfortunately, it has been pointed out by Chen et al. that this is not the case [32]. The basic idea of Chen et al. is to show that for any two-body frustration-free qubit Hamiltonian , there always exists a ground state which is a product of single-or two-qubit states. ...
Measurement-based quantum computing (MBQC) is a model of quantum computing that proceeds by sequential measurements of individual spins in an entangled resource state. However, it remains a challenge to produce efficiently such resource states. Would it be possible to generate these states by simply cooling a quantum many-body system to its ground state? Cluster states, the canonical resource states for MBQC, do not occur naturally as unique ground states of physical systems. This inherent hurdle has led to a significant effort to identify alternative resource states that appear as ground states in spin lattices. Recently, some interesting candidates have been identified with various valence-bond-solid (VBS) states. In this review, we provide a pedagogical introduction to recent progress regarding MBQC with VBS states as possible resource states. This study has led to an interesting interdisciplinary research area at the interface of quantum information science and condensed matter physics.
