Non-Threshold Quantum Secret Sharing Schemes in the Graph State Formalism

In a recent work, Markham and Sanders have proposed a framework to study quantum secret sharing (QSS) schemes using graph states. This framework unified three classes of QSS protocols, namely, sharing classical secrets over private and public channels, and sharing quantum secrets. However, most work on secret sharing based on graph states focused on threshold schemes. In this paper, we focus on general access structures. We show how to realize a large class of arbitrary access structures using the graph state formalism. We show an equivalence between $[[n,1]]$ binary quantum codes and graph state secret sharing schemes sharing one bit. We also establish a similar (but restricted) equivalence between a class of $[[n,1]]$ Calderbank-Shor-Steane (CSS) codes and graph state QSS schemes sharing one qubit. With these results we are able to construct a large class of quantum secret sharing schemes with arbitrary access structures.Comment: LaTeX, 6 page

Scheme for constructing graphs associated with stabilizer quantum codes

We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.Comment: 42 pages, 12 figure

Quantum Copy-Protection and Quantum Money

Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner's scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copy-protected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publicly-verifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its input-output behavior can be quantumly copy-protected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a "Complexity-Theoretic No-Cloning Theorem," which generalizes both the standard No-Cloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum t-designs. Moving beyond the oracle world, we also present an explicit candidate scheme for publicly-verifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copy-protecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption.)Comment: 14-page conference abstract; full version hasn't appeared and will never appear. Being posted to arXiv mostly for archaeological purposes. Explicit money scheme has since been broken by Lutomirski et al (arXiv:0912.3825). Other quantum money material has been superseded by results of Aaronson and Christiano (coming soon). Quantum copy-protection ideas will hopefully be developed in separate wor

Classical simulations of Abelian-group normalizer circuits with intermediate measurements

Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits [arXiv:1201.4867]: a normalizer circuit over a finite Abelian group $G$ is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In [arXiv:1201.4867] it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations [quant-ph/9705052, quant-ph/9807006] to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism [quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.Comment: 26 pages+appendices. Title has changed in this second version. To appear in Quantum Information and Computation, Vol.14 No.3&4, 201

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

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 by Almheiri et. al in arXiv:1411.7041.Comment: 40 Pages + 25 Pages of Appendices. 38 figures. Typos and bibliographic amendments and minor correction