Starting from the one-way group action framework of Brassard and Yung (Crypto 90), we revisit building cryptography based on group actions. Several previous candidates for one-way group actions no longer stand, due to progress both on classical algor
ithms (e.g., graph isomorphism) and quantum algorithms (e.g., discrete logarithm). We propose the general linear group action on tensors as a new candidate to build cryptography based on group actions. Recent works (Futorny--Grochow--Sergeichuk, Lin. Alg. Appl., 2019) suggest that the underlying algorithmic problem, the tensor isomorphism problem, is the hardest one among several isomorphism testing problems arising from areas including coding theory, computational group theory, and multivariate cryptography. We present evidence to justify the viability of this proposal from comprehensive study of the state-of-art heuristic algorithms, theoretical algorithms, and hardness results, as well as quantum algorithms. We then introduce a new notion called pseudorandom group actions to further develop group-action based cryptography. Briefly speaking, given a group $G$ acting on a set $S$, we assume that it is hard to distinguish two distributions of $(s, t)$ either uniformly chosen from $Stimes S$, or where $s$ is randomly chosen from $S$ and $t$ is the result of applying a random group action of $gin G$ on $s$. This subsumes the classical decisional Diffie-Hellman assumption when specialized to a particular group action. We carefully analyze various attack strategies that support the general linear group action on tensors as a candidate for this assumption. Finally, we establish the quantum security of several cryptographic primitives based on the one-way group action assumption and the pseudorandom group action assumption.
We show that any language in nondeterministic time $exp(exp(cdots exp(n)))$, where the number of iterated exponentials is an arbitrary function $R(n)$, can be decided by a multiprover interactive proof system with a classical polynomial-time verifier
and a constant number of quantum entangled provers, with completeness $1$ and soundness $1 - exp(-Cexp(cdotsexp(n)))$, where the number of iterated exponentials is $R(n)-1$ and $C>0$ is a universal constant. The result was previously known for $R=1$ and $R=2$; we obtain it for any time-constructible function $R$. The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC17). As a separate consequence of this technique we obtain a different proof of Slofstras recent result (unpublished) on the uncomputability of the entangled value of multiprover games. Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelsons problem on the relation between the commuting operator and tensor product models for quantum correlations.
We propose the concept of pseudorandom states and study their constructions, properties, and applications. Under the assumption that quantum-secure one-way functions exist, we present concrete and efficient constructions of pseudorandom states. The n
on-cloning theorem plays a central role in our study---it motivates the proper definition and characterizes one of the important properties of pseudorandom quantum states. Namely, there is no efficient quantum algorithm that can create more copies of the state from a given number of pseudorandom states. As the main application, we prove that any family of pseudorandom states naturally gives rise to a private-key quantum money scheme.
We present a protocol that transforms any quantum multi-prover interactive proof into a nonlocal game in which questions consist of logarithmic number of bits and answers of constant number of bits. As a corollary, this proves that the promise proble
m corresponding to the approximation of the nonlocal value to inverse polynomial accuracy is complete for QMIP*, and therefore NEXP-hard. This establishes that nonlocal games are provably harder than classical games without any complexity theory assumptions. Our result also indicates that gap amplification for nonlocal games may be impossible in general and provides a negative evidence for the possibility of the gap amplification approach to the multi-prover variant of the quantum PCP conjecture.
We discuss quantum capacities for two types of entanglement networks: $mathcal{Q}$ for the quantum repeater network with free classical communication, and $mathcal{R}$ for the tensor network as the rank of the linear operation represented by the tens
or network. We find that $mathcal{Q}$ always equals $mathcal{R}$ in the regularized case for the samenetwork graph. However, the relationships between the corresponding one-shot capacities $mathcal{Q}_1$ and $mathcal{R}_1$ are more complicated, and the min-cut upper bound is in general not achievable. We show that the tensor network can be viewed as a stochastic protocol with the quantum repeater network, such that $mathcal{R}_1$ is a natural upper bound of $mathcal{Q}_1$. We analyze the possible gap between $mathcal{R}_1$ and $mathcal{Q}_1$ for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network.
We present a classical interactive protocol that verifies the validity of a quantum witness state for the local Hamiltonian problem. It follows from this protocol that approximating the non-local value of a multi-player one-round game to inverse poly
nomial precision is QMA-hard. Our work makes an interesting connection between the theory of QMA-completeness and Hamiltonian complexity on one hand and the study of non-local games and Bell inequalities on the other.
A binary constraint system game is a two-player one-round non-local game defined by a system of Boolean constraints. The game has a perfect quantum strategy if and only if the constraint system has a quantum satisfying assignment [R. Cleve and R. Mit
tal, arXiv:1209.2729]. We show that several concepts including the quantum chromatic number and the Kochen-Specker sets that arose from different contexts fit naturally in the binary constraint system framework. The structure and complexity of the quantum satisfiability problems for these constraint systems are investigated. Combined with a new construct called the commutativity gadget for each problem, several classic NP-hardness reductions are lifted to their corresponding quant
Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interest
ed in local Hamiltonians with certain interaction patterns, such as nearest neighbour interactions on some type of lattices. A necessary condition for a space $V$ to be the ground-state space of some local Hamiltonian with a given interaction pattern, is that the maximally mixed state supported on $V$ is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have $V$ satisfying the necessary condition mentioned above as its ground-state space, by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles.
The problem of finding the ground state of a frustration-free Hamiltonian carrying only two-body interactions between qubits is known to be solvable in polynomial time. It is also shown recently that, for any such Hamiltonian, there is always a groun
d state that is a product of single- or two-qubit states. However, it remains unclear whether the whole ground space is of any succinct structure. Here, we give a complete characterization of the ground space of any two-body frustration-free Hamiltonian of qubits. Namely, it is a span of tree tensor network states of the same tree structure. This characterization allows us to show that the problem of determining the ground state degeneracy is as hard as, but no harder than, its classical analog.
Measurement based quantum computation (MBQC), which requires only single particle measurements on a universal resource state to achieve the full power of quantum computing, has been recognized as one of the most promising models for the physical real
ization of quantum computers. Despite considerable progress in the last decade, it remains a great challenge to search for new universal resource states with naturally occurring Hamiltonians, and to better understand the entanglement structure of these kinds of states. Here we show that most of the resource states currently known can be reduced to the cluster state, the first known universal resource state, via adaptive local measurements at a constant cost. This new quantum state reduction scheme provides simpler proofs of universality of resource states and opens up plenty of space to the search of new resource states, including an example based on the one-parameter deformation of the AKLT state studied in [Commun. Math. Phys. 144, 443 (1992)] by M. Fannes et al. about twenty years ago.
