Ran Raz

We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary $n\times n$ contraction matrix $A$, and a parameter $T \leq \mathrm{poly}(n)$ and outputs the entries of $A^T$, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements. We show various implications and applications of this result: First, we use this algorithm to show that the class of quantum logspace algorithms with only quantum memory and with intermediate measurements is equivalent to the class of quantum logspace algorithms with only quantum memory without intermediate measurements. This shows that the deferred-measurement principle, a fundamental principle of quantum computing, applies also for quantum logspace algorithms (without classical memory). More generally, we give a quantum algorithm with space $O(S + \log T)$ that takes as an input the description of a quantum algorithm with quantum space $S$ and time $T$, with intermediate measurements (without classical memory), and simulates it unitarily with polynomially small error, without intermediate measurements. Since unitary transformations are reversible (while measurements are irreversible) an interesting aspect of this result is that it shows that any quantum logspace algorithm (without classical memory) can be simulated by a reversible quantum logspace algorithm. This proves a quantum analogue of the result of Lange, McKenzie and Tapp that deterministic logspace is equal to reversible logspace. Finally, we use our results to show non-trivial classical simulations of quantum logspace learning algorithms.

Abstract We prove that parallel repetition of the (3-player) GHZ game reduces the value of the game polynomially fast to~0. That is, the value of the GHZ game repeated in parallel $t$ times is at most $t^{-\Omega(1)}$. Previously, only a bound of $\approx \frac{1}{\alpha(t)}$, where $\alpha$ is the inverse Ackermann function, was known (Verbitsky '96). The GHZ game was recently identified by Dinur, Harsha, Venkat and Yuen as a multi-player game where all existing techniques for proving strong bounds on the value of the parallel repetition of the game fail. Indeed, to prove our result we use a completely new proof technique. Dinur, Harsha, Venkat and Yuen speculated that progress on bounding the value of the parallel repetition of the GHZ game may lead to further progress on the general question of parallel repetition of multi-player games. They suggested that the strong correlations present in the GHZ question distribution represent the ``hardest instance'' of the multi-player parallel repetition problem. Another motivation for studying the parallel repetition of the GHZ game comes from the field of quantum information. The GHZ game, first introduced by Greenberger, Horne and Zeilinger, is a central game in the study of quantum entanglement and has been studied in numerous works. For example, it is used for testing quantum entanglement and for device-independent quantum cryptography. In such applications a game is typically repeated to reduce the probability of error, and hence bounds on the value of the parallel repetition of the game may be useful.

I will talk about recent lower bounds for multi-linear arithmetic formulas. A surprising and beautiful application of these bounds in the field of quantum computation was recently found by Scott Aaronson [Scott Aaronson, "Multilinear Formulas and Skepticism of Quantum Computing"], and will be discussed in his talk. In my talk, I plan to give a short introduction to the problem as well as the main ideas of the lower bounds proof. Some more details are given below: Arithmetic formulas for computing the determinant and the permanent of a matrix have been studied since the 19th century. Are there polynomial size formulas for these functions? Although the determinant and the permanent are among the most extensively studied computational problems, polynomial size formulas for these functions are not known. An outstanding open problem in complexity theory is to prove that polynomial size formulas for these functions do not exist. Note, however, that super-polynomial lower bounds for the size of arithmetic formulas are not known for any explicit function and that questions of this type are considered to be among the most challenging open problems in theoretical computer science. An arithmetic formula is multi-linear if the polynomial computed by each of its sub-formulas is multi-linear, that is, in each of its monomials the power of every input variable is at most one. Multi-linear formulas are restricted, as they do not allow the intermediate use of higher powers of variables in order to finally compute a certain multi-linear function. Note, however, that for many multi-linear functions, formulas that are not multi-linear are very counter-intuitive. Note also that both the determinant and the permanent are multi-linear functions and that many of the well known formulas for these functions are multi-linear formulas. We prove that any multi-linear arithmetic formula for the determinant or the permanent of an n dimensional matrix is of size super-polynomial in n. Previously, no lower bound was known (for any explicit function) even for the special case of multi-linear formulas of constant depth.