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A Full Characterization of Quantum Advice Scott Aaronson Andrew Drucker.

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1 A Full Characterization of Quantum Advice Scott Aaronson Andrew Drucker

2 Freeze-Dried Computation Motivating Question: How much useful computational work can one store in a quantum state, for later retrieval? If quantum states are exponentially large objects, then possibly a huge amount! Yet we also know, from Holevos Theorem, that quantum states have no more general-purpose storage capacity than classical strings of the same size

3 Cast of Characters BQP/qpoly is the class of problems solvable in quantum polynomial time, with the help of polynomial-size quantum advice states Formally: a language L is in BQP/qpoly if there exists a polynomial time quantum algorithm A, as well as quantum advice states {| n } n on poly(n) qubits, such that for every input x of size n, A(x,| n ) decides whether or not x L with error probability at most 1/3 YQP (Yoda Quantum Polynomial-Time) is the same, except we also require that for every alleged advice state, A(x, ) outputs either the right answer or FAIL with probability at least 2/3 BQP YQP QMA BQP/qpoly

4 Watrous 2000: For any fixed, finite black-box group G n and subgroup H n G n, deciding membership in H n is in BQP/qpoly The quantum advice state is just an equal superposition |H n over the elements of H n We dont know how to solve the same problem in BQP/poly A. 2004: BQP/qpoly PostBQP/poly P #P /poly Quantum advice can be simulated by classical advice, combined with postselection on unlikely measurement outcomes A. 2006: HeurBQP/qpoly = HeurYQP/poly Trusted quantum advice can be simulated on most inputs by trusted classical advice combined with untrusted quantum advice A.-Kuperberg 2007: There exists a quantum oracle separating BQP/qpoly from BQP/poly Q UANTUM ADVICE IS POWERFUL N O I T I SN T

5 New Result: BQP/qpoly = YQP/poly Trusted quantum advice is equivalent in power to trusted classical advice combined with untrusted quantum advice. (Quantum states never need to be trusted) Given any n-qubit state, there exists a local Hamiltonian H (indeed, a sum of 2D nearest-neighbor interactions) such that: For any ground state | of H, and measuring circuit E with m gates, theres an efficient measuring circuit E such that P HYSICS I MPLICATION : Furthermore, H is on poly(n,m,1/ ) qubits.

6 Implication for Quantum Communication Given any n-qubit state, Alice can send a poly(n)-qubit state and a string x to Bob, in such a way that: can be used to simulate on all small circuits, and Bob can efficiently verify that using x, x

7 Majority- Certificates Lemma Real Majority- Certificates Lemma Circuit Learning (Bshouty et al.) Minimax Theorem Safe Winnowing Lemma Holevos Theorem Random Access Code Lower Bound (Ambainis et al.) BQP/qpoly=YQP/poly HeurBQP/qpoly=HeurYQP/poly (A.06) Quantum advice no harder than ground state preparation Fat-Shattering Bound (A.06) Covering Lemma (Alon et al.) Learning of p- Concept Classes (Bartlett & Long) L OCAL H AMILTONIANS is QMA-complete (Kitaev) Cook-Levin Theorem QMA=QMA+ (Aharonov & Regev) Used as lemma Generalizes

8 Main Tool: Majority-Certificates Lemma (Related to boosting in computational learning theory) Lemma: Let S be a set of Boolean functions f:{0,1} n {0,1}, and let f * S. Then there exist m=O(n) certificates C 1,…,C m, each of size k=O(log|S|), such that (i)Theres a unique f i S consistent with each C i, and (ii)f*(x)=MAJORITY(f 1 (x),…,f m (x)) for all x {0,1} n. Definitions: A certificate is a partial Boolean function C:{0,1} n {0,1,*}. A Boolean function f:{0,1} n {0,1} is consistent with C, if f(x)=C(x) whenever C(x) {0,1}. The size of C is the number of inputs x such that C(x) {0,1}.

9 that computes some Boolean function f:{0,1} n {0,1} belonging to a small set S (meaning, of size 2 poly(n) ). Someone wants to prove to us that f equals (say) the all-0 function, by having us check a polynomial number of outputs f(x 1 ),…,f(x m ). Intuition: Were given a black box (think: quantum state) f xf(x) This is trivially impossible! f0f0 f1f1 f2f2 f3f3 f4f4 f5f5 x1x1 010000 x2x2 001000 x3x3 000100 x4x4 000010 x5x5 000001 But … what if we get 3 black boxes, and are allowed to simulate f=f 0 by taking the point-wise MAJORITY of their outputs?

10 Lifting the Lemma to Quantumland Boolean Majority-CertificatesBQP/qpoly=YQP/poly Proof Set S of Boolean functionsSet S of p(n)-qubit mixed states True function f * STrue advice state | n Other functions f 1,…,f m Other states 1,…, m Certificate C i to isolate f i Measurement E i to isolate I New DifficultySolution The class of p(n)-qubit quantum states is infinitely large! And even if we discretize it, its still doubly-exponentially large Result of A.06 on learnability of quantum states (building on Ambainis et al. 1999) Instead of Boolean functions f:{0,1} n {0,1}, now we have real functions f :{0,1} n [0,1] representing the expectation values Learning theory has tools to deal with this: fat-shattering dimension, -covers… (Alon et al. 1997) How do we verify a quantum witness without destroying it? QMA=QMA+ (Aharonov & Regev 2003) What if a certificate asks us to verify Tr(E )a, but Tr(E ) is right at the knife-edge? Safe Winnowing Lemma

11 Quantum Karp-Lipton Theorem: An Unexpected Application of Our BQP/qpoly=YQP/poly Theorem Our quantum analogue: If NP BQP/qpoly, then coNP NP QMA PromiseQMA. Karp-Lipton 1982: If NP P/poly, then coNP NP = NP NP. Idea: Let M be a YQP/poly machine that solves 3SAT. In QMA, guess the classical advice z to M, and check that some quantum witness | is consistent with z. Then, in PromiseQMA, search for a quantum witness | consistent with z, as well as a 3SAT instance of size n on which | fails. If no such instance is found, guess the first quantified string of the coNP NP statement, and use | to find the second quantified string.

12 Open Problems Does QMA=QCMA? Does BQP/qpoly=BQP/poly? Can we at least prove (classical) oracle separations? Improve the parameters of the majority-certificates lemma, and clarify the connection with boosting? Other applications of majority-certificates? Is it possible that every state on n qubits can be simulated by a verifiable state on n qubits, rather than poly(n)?

13 If you can make the following terms comprehensible to a computer scientist: Squeezed state Parametric downconversion Homodyne measurement please see me after the talk

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