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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