Quantum Information and the PCP Theorem

We show how to encode $2^n$ (classical) bits $a_1,...,a_{2^n}$ by a single quantum state $|\Psi>$ of size O(n) qubits, such that: for any constant $k$ and any $i_1,...,i_k \in \{1,...,2^n\}$, the values of the bits $a_{i_1},...,a_{i_k}$ can be retrieved from $|\Psi>$ by a one-round Arthur-Merlin interactive protocol of size polynomial in $n$. This shows how to go around Holevo-Nayak's Theorem, using Arthur-Merlin proofs. We use the new representation to prove the following results: 1) Interactive proofs with quantum advice: We show that the class $QIP/qpoly$ contains ALL languages. That is, for any language $L$ (even non-recursive), the membership $x \in L$ (for $x$ of length $n$) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state $|\Psi_{L,n} >$ (depending only on $L$ and $n$). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2) PCP with only one query: We show that the membership $x \in SAT$ (for $x$ of length $n$) can be proved by a logarithmic-size quantum state $|\Psi >$, together with a polynomial-size classical proof consisting of blocks of length $polylog(n)$ bits each, such that after measuring the state $|\Psi >$ the verifier only needs to read {\bf one} block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum low-degree-test that may be interesting in its own right.Comment: 30 page

Oracles Are Subtle But Not Malicious

Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new lower bound for perceptrons and low- degree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first nonrelativizing separation of "traditional" complexity classes, as opposed to interactive proof classes such as MIP and MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size n^k for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size n^k with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand, we also show that the NP queries could be parallelized if P=NP. Thus, classes such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish between relativizing and black-box techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem.Comment: 20 pages, 1 figur

Shadow Tomography of Quantum States

We introduce the problem of *shadow tomography*: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability that $E_{i}$ accepts $\rho$, to within additive error $\varepsilon$, for each of the $M$ measurements. How many copies of $\rho$ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only $\widetilde{O}\left( \varepsilon^{-4}\cdot\log^{4} M\cdot\log D\right)$ copies. This means, for example, that we can learn the behavior of an arbitrary $n$-qubit state, on all accepting/rejecting circuits of some fixed polynomial size, by measuring only $n^{O\left( 1\right)}$ copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Brand\~ao et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.Comment: 29 pages, extended abstract appeared in Proceedings of STOC'2018, revised to give slightly better upper bound (1/eps^4 rather than 1/eps^5) and lower bounds with explicit dependence on the dimension

Two Results about Quantum Messages

We show two results about the relationship between quantum and classical messages. Our first contribution is to show how to replace a quantum message in a one-way communication protocol by a deterministic message, establishing that for all partial Boolean functions $f:\{0,1\}^n\times\{0,1\}^m\to\{0,1\}$ we have $D^{A\to B}(f)\leq O(Q^{A\to B,*}(f)\cdot m)$. This bound was previously known for total functions, while for partial functions this improves on results by Aaronson, in which either a log-factor on the right hand is present, or the left hand side is $R^{A\to B}(f)$, and in which also no entanglement is allowed. In our second contribution we investigate the power of quantum proofs over classical proofs. We give the first example of a scenario, where quantum proofs lead to exponential savings in computing a Boolean function. The previously only known separation between the power of quantum and classical proofs is in a setting where the input is also quantum. We exhibit a partial Boolean function $f$, such that there is a one-way quantum communication protocol receiving a quantum proof (i.e., a protocol of type QMA) that has cost $O(\log n)$ for $f$, whereas every one-way quantum protocol for $f$ receiving a classical proof (protocol of type QCMA) requires communication $\Omega(\sqrt n/\log n)$