Download presentation

Presentation is loading. Please wait.

Published bySarah Stone Modified over 4 years ago

1
The Power 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 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 PP/poly = PostBQP/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 an n-qubit state and parameters m,, there exists a local Hamiltonian H on poly(n,m,1/ ) qubits (e.g., a sum of 2- qubit interactions) for which the following holds: For any ground state | of H, and any binary measurement E on performed by a circuit with m gates, theres an efficient measurement f(E) that we can perform on | such that F OR THE P HYSICISTS

6
What Does It Mean? Preparing quantum advice states is no harder than preparing ground states of local Hamiltonians This explains a once-mysterious relationship between quantum proofs and quantum advice: efficient preparability of ground states would imply both QMA=QCMA and BQP/qpoly=BQP/poly Quantum Karp-Lipton Theorem: NP-complete problems are not efficiently solvable using quantum advice, unless some uniform complexity classes collapse unexpectedly QCMA/qpoly QMA/poly: classical proofs and quantum advice can be simulated with quantum proofs and classical advice

7
BQP YQPQCMABQP/poly BQP/qpoly =YQP/poly QCMA/polyQMA QCMA/qpoly QMA/polyPP PP/polyQMA/qpoly PSPACE/poly A.06 This work

8
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

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
Majority-Certificates Lemma 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)Some f i S is consistent with each C i, and (ii)If f i S is consistent with C i for all i, then MAJ(f 1 (x),…,f m (x))=f * (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}.

11
Proof Idea By symmetry, we can assume f * is the all-0 function. Consider a two-player, zero-sum matrix game: Alice picks a certificate C of size k consistent with some f S Bob picks an input x {0,1} n Alice wins this game if f(x)=0 for all f S consistent with C. Crucial Claim: Alice has a mixed strategy that lets her win >90% of the time. The lemma follows from this claim! Just choose certificates C 1,…,C m independently from Alices winning distribution. Then by a Chernoff bound, almost certainly MAJ(f 1 (x),…,f m (x))=0 for all f 1,…,f m consistent with C 1,…,C m respectively and all inputs x {0,1} n. So clearly there exist C 1,…,C m with this property.

12
Proof of Claim Use the Minimax Theorem! Given a distribution D over x, its enough to create a fixed certificate C such that Stage I: Choose x 1,…,x t independently from D, for some t=O(log|S|). Then with high probability, requiring f(x 1 )=…=f(x t )=0 kills off every f S such that Stage II: Repeatedly add a constraint f(x i )=b i that kills at least half the remaining functions. After log 2 |S| iterations, well have winnowed S down to just a single function f S.

13
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

14
Theorem: BQP/qpoly = YQP/poly. Proof Sketch: YQP/poly BQP/qpoly is immediate. For the other direction, let L BQP/qpoly. Let M be a quantum algorithm that decides L using advice state | n. Define Let S = {f : }. Then S has fat-shattering dimension at most poly(n), by A.06. So we can apply a real analogue of the Majority-Certificates Lemma to S. This yields certificates C 1,…,C m (for some m=poly(n)), such that any states 1,…, m consistent with C 1,…,C m respectively satisfy for all x {0,1} n (regardless of entanglement). To check the C i s, we use the QMA+ super-verifier of Aharonov & Regev.

15
Promised Application to Physics Furthermore, in their reduction, the witness is a history state So given any language L BQP/qpoly=YQP/poly, we can use the Kitaev et al. reduction to get a local Hamiltonian H whose unique ground state is |. We can then use | to recover the YQP witness |, and thereby decide L By Kitaev et al., we know L OCAL H AMILTONIANS is QMA-complete. Measuring this state yields the original QMA witness | 1 with (1/poly(n)) probability. Hence | 1 can be recovered from

16
Quantum Karp-Lipton 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. Proof Idea: A coNP NP statement has the form x y R(x,y). By the hypothesis and BQP/qpoly = YQP/poly, there exists an advice string s, such that any quantum state consistent with s lets us solve NP problems (and some such is consistent). In QMA PromiseQMA, first guess an s thats consistent with some state. Then use the oracle to search for an x and such that, if is consistent with s, then R(x,Q(x, )) holds, where Q is a quantum algorithm that searches for a y such that R(x,y).

17
A Theory of Isolatability Which classes of functions C are isolatablein the sense that for any f C, one can give a small number of conditions such that any f 1,…,f m C satisfying the conditions can be used to compute f efficiently on all inputs? We can generalize the majority-certificates idea well beyond what we have any application for Another application of the Majority-Certificates Lemma: it substantially simplifies the proof that BQPSPACE/coin = PSPACE/poly We study the following abstract question, inspired by computational learning theory:

18
Open Problems Improve QMA/qpoly PSPACE/poly to QMA/qpoly P #P /poly Find other applications of the majority-certificates technique Circuit complexity? Communication complexity? Learning theory? Quantum information? Is the dependence on n, log|S|, and 1/ optimal? Prove a classical oracle separation between BQP/poly and BQP/qpoly=YQP/poly Although this work closes off a chapter in the quantum advice story, there are still

Similar presentations

OK

BQP/qpoly EXP/poly Scott Aaronson UC Berkeley. BQP/qpoly Class of languages recognized by a bounded-error polytime quantum algorithm, with a polysize.

BQP/qpoly EXP/poly Scott Aaronson UC Berkeley. BQP/qpoly Class of languages recognized by a bounded-error polytime quantum algorithm, with a polysize.

© 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 cross docking operations Ppt on thermal power generation Ppt on time management for engineering students Ppt on index numbers page Ppt on sweat equity shares Ppt on online library management Ppt on icici bank history Ppt on surface tension of water Ppt on waxes chemical structure Ppt on natural and human nature