Download presentation

Presentation is loading. Please wait.

Published byJacob Whitaker Modified over 4 years ago

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

Similar presentations

OK

How to Solve Longstanding Open Problems In Quantum Computing Using Only Fourier Analysis Scott Aaronson (MIT) For those who hate quantum: The open problems.

How to Solve Longstanding Open Problems In Quantum Computing Using Only Fourier Analysis Scott Aaronson (MIT) For those who hate quantum: The open problems.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on law against child marriage bangladesh Ppt on power diode Ppt on diode as rectifier regulator Ppt on computer languages to learn Ppt on power generation by speed breaker ahead Ppt on lathe machine parts Ppt on air pollution act Ppt on waves tides and ocean currents and climate Ppt on surface water drains A ppt on positive thinking