We 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 Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].

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 31 P impurities in an almost isotopically pure sample of 28 Si 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 ( 13 C) 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.

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 statedependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form suð1; 1Þ 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. DOI: 10.1103/PhysRevLett.120.121602 Introduction.-Applications of quantum information concepts to high-energy physics and gravity have recently led to many far-reaching developments. In particular, it has become apparent that special properties of entanglement in holographic [1] quantum field theory (QFT) states are crucial for the emergence of smooth higher-dimensional (bulk) geometries in the gauge-gravity duality [2]. Much of the progress in this direction was achieved by building on the holographic entanglement entropy proposal by Ryu and Takayanagi [3], which geometrizes the von Neumann entropy of a reduced density matrix of a QFT in a subregion in terms of the area of codimension-2 bulk minimal surfaces anchored at the boundary of this subregion (see, e.g., Ref. [4] for a recent overview). However, Ryu-Takayanagi surfaces are often unable to access the whole holographic geometry [5][6][7]. This observation has led to significant interest in novel, from the point of view of quantum gravity, codimension-1 (volume) and codimension-0 (action) bulk quantities, whose behavior suggests conjecturing a link with the information theory notion of quantum state complexity [8][9][10][11][12][13][14]. In fact, a certain identification between complexity and action was originally suggested by Toffoli [15,16] outside the context of holography.

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 König. 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.