A Full Characterization of Quantum Advice

Scott AaronsonAndrew Drucker

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 Holevo’s Theorem, that quantum states have no more “general-purpose storage capacity” than classical strings of the same size

Cast of CharactersBQP/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 xL 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

Watrous 2000: For any fixed, finite black-box group Gn and subgroup Hn≤Gn, deciding membership in Hn is in BQP/qpoly

The quantum advice state is just an equal superposition |Hn over the elements of Hn We don’t 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/polyTrusted 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

QUANTUM ADVICE IS POWERFUL

NO IT ISN’T

New Result: BQP/qpoly = YQP/polyTrusted 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, there’s an efficient measuring circuit E’ such that

.Tr' EE

“PHYSICS” IMPLICATION:

Furthermore, H is on poly(n,m,1/) qubits.

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

Majority-Certificates

Lemma

Real Majority-Certificates Lemma

Circuit Learning (Bshouty et al.)

Minimax Theorem

Safe Winnowing

Lemma

Holevo’s 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)

LOCAL HAMILTONIANS is QMA-complete

(Kitaev)

Cook-Levin Theorem

QMA=QMA+(Aharonov & Regev)

Used as lemma

Generalizes

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 C1,…,Cm, each of size k=O(log|S|), such that

(i)There’s a unique fiS consistent with each Ci, and

(ii)f*(x)=MAJORITY(f1(x),…,fm(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}.

that computes some Boolean function f:{0,1}n{0,1} belonging to a “small” set S (meaning, of size 2poly(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(x1),…,f(xm).

Intuition: We’re given a black box (think: quantum state)

fx f(x)

This is trivially impossible!f0 f1 f2 f3 f4 f5

x1 0 1 0 0 0 0

x2 0 0 1 0 0 0

x3 0 0 0 1 0 0

x4 0 0 0 0 1 0

x5 0 0 0 0 0 1

But … what if we get 3 black boxes, and are allowed to simulate f=f0 by taking the point-wise MAJORITY of their outputs?

“Lifting” the Lemma to QuantumlandBoolean Majority-Certificates BQP/qpoly=YQP/poly Proof

Set S of Boolean functions Set S of p(n)-qubit mixed states

“True” function f*S “True” advice state |n

Other functions f1,…,fm Other states 1,…,m

Certificate Ci to isolate fi Measurement Ei to isolate I

New Difficulty Solution

The class of p(n)-qubit quantum states is infinitely large! And even if we discretize it, it’s 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”

Quantum Karp-Lipton Theorem:An Unexpected Application of Our BQP/qpoly=YQP/poly Theorem

Our quantum analogue:

If NP BQP/qpoly, then coNPNP QMAPromiseQMA.

Karp-Lipton 1982: If NP P/poly, then coNPNP = NPNP.

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 coNPNP statement, and use | to find the second quantified string.

Open ProblemsDoes 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)?

If you can make the following terms comprehensible to a computer scientist:

“Squeezed state”

“Parametric downconversion”

“Homodyne measurement”

please see me after the talk