Fernando Pastawski

Room-Temperature Quantum Bit Memory Exceeding One Second

Extending Quantum Memory Practical applications in quantum communication and quantum computation require the building blocks—quantum bits and quantum memory—to be sufficiently robust and long-lived to allow for manipulation and storage (see the Perspective by Boehme and McCarney). Steger et al. (p. 1280) demonstrate that the nuclear spins of 31P impurities in an almost isotopically pure sample of 28Si can have a coherence time of as long as 192 seconds at a temperature of ∼1.7 K. In diamond at room temperature, Maurer et al. (p. 1283) show that a spin-based qubit system comprised of an isotopic impurity (13C) in the vicinity of a color defect (a nitrogen-vacancy center) could be manipulated to have a coherence time exceeding one second. Such lifetimes promise to make spin-based architectures feasible building blocks for quantum information science. Defects in diamond can be operated as quantum memories at room temperature. Stable quantum bits, capable both of storing quantum information for macroscopic time scales and of integration inside small portable devices, are an essential building block for an array of potential applications. We demonstrate high-fidelity control of a solid-state qubit, which preserves its polarization for several minutes and features coherence lifetimes exceeding 1 second at room temperature. The qubit consists of a single 13C nuclear spin in the vicinity of a nitrogen-vacancy color center within an isotopically purified diamond crystal. The long qubit memory time was achieved via a technique involving dissipative decoupling of the single nuclear spin from its local environment. The versatility, robustness, and potential scalability of this system may allow for new applications in quantum information science.

Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence

A bstractWe propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].

Quantum memories based on engineered dissipation

Storing quantum information for long times without disruptions is a major requirement for most quantum information technologies. A very appealing approach is to use self-correcting Hamiltonians, that is tailoring local interactions among the qubits such that when the system is weakly coupled to a cold bath the thermalization process takes a long time. Here we propose an alternative but more powerful approach in which the coupling to a bath is engineered, so that dissipation protects the encoded qubit against more general kinds of errors. We show that the method can be implemented locally in four-dimensional lattice geometries by means of a toric code and propose a simple two-dimensional setup for proof-of-principle experiments.

Toward a Definition of Complexity for Quantum Field Theory States

We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form su(1,1) algebras. On the manifold of Gaussian states generated by these operations, the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and those of holographic complexity proposals.

Unforgeable Noise-Tolerant Quantum Tokens

The realization of devices that harness the laws of quantum mechanics represents an exciting challenge at the interface of modern technology and fundamental science. An exemplary paragon of the power of such quantum primitives is the concept of “quantum money” [Wiesner S (1983) ACM SIGACT News 15:78–88]. A dishonest holder of a quantum bank note will invariably fail in any counterfeiting attempts; indeed, under assumptions of ideal measurements and decoherence-free memories such security is guaranteed by the no-cloning theorem. In any practical situation, however, noise, decoherence, and operational imperfections abound. Thus, the development of secure “quantum money”-type primitives capable of tolerating realistic infidelities is of both practical and fundamental importance. Here, we propose a novel class of such protocols and demonstrate their tolerance to noise; moreover, we prove their rigorous security by determining tight fidelity thresholds. Our proposed protocols require only the ability to prepare, store, and measure single quantum bit memories, making their experimental realization accessible with current technologies.

Unfolding the color code

The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a d-dimensional closed manifold is equivalent to multiple decoupled copies of the d-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for d = 2, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for the d-dimensional color code with d + 1 boundaries of d + 1 distinct colors, we find that the code is equivalent to multiple copies of the d-dimensional toric code which are attached along a (d - 1)-dimensional boundary. In particular, for d = 2, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the d-dimensional toric code admits logical non-Pauli gates from the dth level of the Clifford hierarchy, and thus saturates the bound by Bravyi and Konig. In particular, we show that the logical d-qubit control-Z gate can be fault-tolerantly implemented on the stack of d copies of the toric code by a local unitary transformation.

Hypercontractivity of quasi-free quantum semigroups

Hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate. A logarithmic Sobolev inequality and the corresponding logarithmic Sobolev constant can be inferred from the semigroupʼs hypercontractive norm bound. We consider completely-positive quantum mechanical semigroups described by a Lindblad master equation. To prove the norm bound, we follow an approach which has its roots in the study of classical rate equations. We use interpolation theorems for non-commutative spaces to obtain a general hypercontractive inequality from a particular -norm bound. Then, we derive a bound on the -norm from an analysis of the block diagonal structure of the semigroupʼs spectrum. We show that the dynamics of an N-qubit graph state Hamiltonian weakly coupled to a thermal environment is hypercontractive. As a consequence this allows for the efficient preparation of graph states in time by coupling at sufficiently low temperature. Furthermore, we extend our results to gapped Liouvillians arising from a weak linear coupling of a free-fermion systems.

Fault-tolerant logical gates in quantum error-correcting codes

Recently, S. Bravyi and R. Konig [Phys. Rev. Lett. 110, 170503 (2013)] have shown that there is a trade-off between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant-depth geometrically local circuit and are thus fault tolerant by construction. In particular, they show that, for local stabilizer codes in D spatial dimensions, locality-preserving gates are restricted to a set of unitary gates known as the Dth level of the Clifford hierarchy. In this paper, we explore this idea further by providing several extensions and applications of their characterization to qubit stabilizer and subsystem codes. First, we present a no-go theorem for self-correcting quantum memory. Namely, we prove that a three-dimensional stabilizer Hamiltonian with a locality-preserving implementation of a non-Clifford gate cannot have a macroscopic energy barrier. This result implies that non-Clifford gates do not admit such implementations in Haahs cubic code and Michnickis welded code. Second, we prove that the code distance of a D-dimensional local stabilizer code with a nontrivial locality-preserving mth-level Clifford logical gate is upper bounded by O(L^(D+1−m)). For codes with non-Clifford gates (m>2), this improves the previous best bound by S. Bravyi and B. Terhal [New. J. Phys. 11, 043029 (2009)]. Topological color codes, introduced by H. Bombin and M. A. Martin-Delgado [Phys. Rev. Lett. 97, 180501 (2006); Phys. Rev. Lett. 98, 160502 (2007); Phys. Rev. B 75, 075103 (2007)], saturate the bound for m=D. Third, we prove that the qubit erasure threshold for codes with a nontrivial transversal mth-level Clifford logical gate is upper bounded by 1/m. This implies that no family of fault-tolerant codes with transversal gates in increasing level of the Clifford hierarchy may exist. This result applies to arbitrary stabilizer and subsystem codes and is not restricted to geometrically local codes. Fourth, we extend the result of Bravyi and Konig to subsystem codes. Unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of an error threshold in a subsystem code, and a conclusion analogous to that of Bravyi and Konig is recovered.

Protected gates for topological quantum field theories

We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators --- for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons; in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.

Selective and efficient quantum process tomography

In this paper we describe in detail and generalize a method for quantum process tomography that was presented by Bendersky et al. [Phys. Rev. Lett. 100, 190403 (2008)]. The method enables the efficient estimation of any element of the