We show how to perform universal adiabatic quantum computation using a Hamiltonian which describes a set of particles with local interactions on a two-dimensional grid. A single parameter in the Hamiltonian is adiabatically changed as a function of time to simulate the quantum circuit. We bound the eigenvalue gap above the unique groundstate by mapping our model onto the ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin chain was computed exactly by Koma and Nachtergaele using its q-deformed version of SU(2) symmetry. We also discuss a related time-independent Hamiltonian which was shown by Janzing to be capable of universal computation. We observe that in the limit of large system size, the time evolution is equivalent to the exactly solvable quantum walk on Young's lattice. [3,4]. Although the class of universal Hamiltonians originally considered (nearest neighbor interactions between six dimensional particles in two dimensions) is not practically viable, perturbation gadget techniques [5,6] were later used to massage it into simpler forms [7,8]. However, these techniques have the disadvantage of requiring impractically high variability in the coupling strengths which appear in the Hamiltonian (see, e.g., the analysis in [9]). Given this state of affairs, it is of interest to consider how to construct a universal adiabatic quantum computer with a simple Hamiltonian without using perturbative gadgets.An alternative type of circuit-to-Hamiltonian mapping which is conceptually distinct from the Feynman-Kitaev construction has been used by some authors [10][11][12][13][14][15][16]. In these works a quantum circuit is mapped to a Hamiltonian which acts on a Hilbert space with computational and "local" clock degrees of freedom associated with every qubit in the circuit. This idea was first explored by Margolus in 1989 [10], just four years after Feynman's celebrated paper on Hamiltonian computation [3]. Margolus showed how to simulate a one-dimensional cellular automaton by Schrödinger time evolution with a time-independent Hamiltonian. More recently, Janzing [11] presented a scheme for universal computation with a time-independent Hamiltonian. In reference [14] it was claimed that an approach along these lines can be used to perform universal adiabatic quantum computation; unfortunately, the analysis presented by Mizel et al. does not establish the claimed results. The local clock idea was developed further in the recent "space-time circuitto-Hamiltonian construction" and was used to prove that approximating the ground energy of a certain class of interacting particle systems is QMA-complete [16].Our main result is a new method which achieves efficient universal adiabatic quantum computation using the space-time circuit-to-Hamiltonian construction. The Hamiltonian we use describes a simple system of interacting particles which live on the edges of a two dimensional grid. To prove that the resulting algorithm is efficient we use a mapping from our Hamiltonian to the ferromagnetic XXZ model...

We prove Lieb-Robinson bounds and the existence of the thermodynamic limit for a general class of irreversible dynamics for quantum lattice systems with time-dependent generators that satisfy a suitable decay condition in space.1991 Mathematics Subject Classification. 82C10, 82C20, 37L60, 46L57.

The Data Processing Inequality (DPI) says that the Umegaki relative entropy S(ρ||σ) := Tr[ρ(log ρ − log σ)] is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let M be a finite dimensional von Neumann algebra and N a von Neumann subalgebra of it. Let E τ be the tracial conditional expectation from M onto N . For density matrices ρ and σ in M, let ρ N := E τ ρ and σ N := E τ σ. Since E τ is CPTP, the DPI says that S(ρ||σ) ≥ S(ρ N ||σ N ), and the general case is readily deduced from this. A theorem of Petz says that there is equality if and only if σ = R ρ (σ N ), where R ρ is the Petz recovery map, which is dual to the Accardi-Cecchini coarse graining operator A ρ from M to N . We prove a quantitative version of Peta's theorem. In it simplest form, our bound iswhere ∆ σ,ρ is the relative modular operator. Since ∆ σ,ρ ≤ ρ −1 , this bound implies a bound with a constant that is independent of σ. We also prove an analogous result with a more complicated constant in which the roles of ρ and σ are interchanged on the right. Explicitly describing the solutions set of the Petz equation σ = R ρ (σ N ) amounts to determining the set of fixed points of the Accardi-Cecchini coarse graining map. Building on previous work, we provide a throughly detailed description of the set of solutions of the Petz equation R ρ (E τ γ) = γ, and obtain all of our results in a simple, self contained manner. Finally, we prove a theorem characterizing state ρ for which the orthogonal projection from M onto N in the GNS inner product is a conditional expectation.1.1 Definition. For any state ρ on M, ρ N denotes the state on N given by ρ N := E τ (ρ) where, as always E τ denotes the tracial conditional expectation onto N .The restriction of a state ρ on M to N is of course a state on N , and as such, it is represented by a unique density matrix belonging to N , which is precisely ρ N .

We prove number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies S f (ρ||σ) defined in terms of an operator monotone decreasing functions f ; and in particular, the Rényi relative entropies. Included in our results are bounds in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras where the results are already non-trivial and of interest in quantum information theory.

We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum state. We show that this inequality also gives rise to several related inequalities whose counterparts are well-known in the classical setting: in particular, it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality, and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup. As an application, we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup, and argue that it implies fast convergence towards the fixed point for a large class of initial states.for the convolution (3) and for λ = 1/2, where S(ρ X ) = −tr(ρ X log ρ X ) denotes the von Neumann entropy. Subsequent work [14] managed to lift the restriction on λ, and generalized this result to more general (Gaussian) unitaries in place of U λ . A related inequality of the form , 1] was also shown in [13], generalizing classical results [15] (see also [16] for a discussion of the relationship between the two). A generalization to conditional entropies was proposed in [17], and an application to channel capacities was discussed in [18].A key tool in establishing these results is the quantum Fisher information J(ρ), defined for a state ρ as the divergence-based Fisher information of the family ρ (θ) θ obtained by displacing ρ along a phase space direction (see Section 3 for a precise definition). It was shown in [13] for λ = 1/2 and in [14] for general λ ∈ [0, 1] that this quantity satisfies the Fisher information inequality [3]