Efficient Quantum Tensor Product Expanders and k-designs

Quantum expanders are a quantum analogue of expanders, and k-tensor product expanders are a generalisation to graphs that randomise k correlated walkers. Here we give an efficient construction of constant-degree, constant-gap quantum k-tensor product expanders. The key ingredients are an efficient classical tensor product expander and the quantum Fourier transform. Our construction works whenever k=O(n/log n), where n is the number of qubits. An immediate corollary of this result is an efficient construction of an approximate unitary k-design, which is a quantum analogue of an approximate k-wise independent function, on n qubits for any k=O(n/log n). Previously, no efficient constructions were known for k>2, while state designs, of which unitary designs are a generalisation, were constructed efficiently in [Ambainis, Emerson 2007].Comment: 16 pages, typo in references fixe

Exponential Quantum Speed-ups are Generic

A central problem in quantum computation is to understand which quantum circuits are useful for exponential speed-ups over classical computation. We address this question in the setting of query complexity and show that for almost any sufficiently long quantum circuit one can construct a black-box problem which is solved by the circuit with a constant number of quantum queries, but which requires exponentially many classical queries, even if the classical machine has the ability to postselect. We prove the result in two steps. In the first, we show that almost any element of an approximate unitary 3-design is useful to solve a certain black-box problem efficiently. The problem is based on a recent oracle construction of Aaronson and gives an exponential separation between quantum and classical bounded-error with postselection query complexities. In the second step, which may be of independent interest, we prove that linear-sized random quantum circuits give an approximate unitary 3-design. The key ingredient in the proof is a technique from quantum many-body theory to lower bound the spectral gap of local quantum Hamiltonians.Comment: 24 pages. v2 minor correction

Efficient Quantum Pseudorandomness

Randomness is both a useful way to model natural systems and a useful tool for engineered systems, e.g. in computation, communication and control. Fully random transformations require exponential time for either classical or quantum systems, but in many case pseudorandom operations can emulate certain properties of truly random ones. Indeed in the classical realm there is by now a well-developed theory of such pseudorandom operations. However the construction of such objects turns out to be much harder in the quantum case. Here we show that random quantum circuits are a powerful source of quantum pseudorandomness. This gives the for the first time a polynomialtime construction of quantum unitary designs, which can replace fully random operations in most applications, and shows that generic quantum dynamics cannot be distinguished from truly random processes. We discuss applications of our result to quantum information science, cryptography and to understanding self-equilibration of closed quantum dynamics.Comment: 6 pages, 1 figure. Short version of http://arxiv.org/abs/1208.069

Variations on Classical and Quantum Extractors

Many constructions of randomness extractors are known to work in the presence of quantum side information, but there also exist extractors which do not [Gavinsky {\it et al.}, STOC'07]. Here we find that spectral extractors $\psi$ with a bound on the second largest eigenvalue $\lambda_{2}(\psi^{\dagger}\circ\psi)$ are quantum-proof. We then discuss fully quantum extractors and call constructions that also work in the presence of quantum correlations decoupling. As in the classical case we show that spectral extractors are decoupling. The drawback of classical and quantum spectral extractors is that they always have a long seed, whereas there exist classical extractors with exponentially smaller seed size. For the quantum case, we show that there exists an extractor with extremely short seed size $d=O(\log(1/\epsilon))$, where $\epsilon>0$ denotes the quality of the randomness. In contrast to the classical case this is independent of the input size and min-entropy and matches the simple lower bound $d\geq\log(1/\epsilon)$.Comment: 7 pages, slightly enhanced IEEE ISIT submission including all the proof