A Full Characterization of Quantum Advice

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 Holevo’s 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. Quantum advice is powerful Watrous 2000: For any fixed, finite black-box group Gn and subgroup Hn≤Gn, deciding membership in Hn is in BQP/qpolyThe 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.-Kuperberg 2007: There exists a “quantum oracle” separating BQP/qpolyfromBQP/poly No It Isn’t 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

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”) “Physics” Implication: 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 Furthermore, H is on poly(n,m,1/) qubits.

6. Implication for Quantum Communication , x 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

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

8. Main Tool: Majority-Certificates Lemma(Related to boosting in computational learning theory) 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}. • 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 • There’s a unique fiS consistent with each Ci, and • f*(x)=MAJORITY(f1(x),…,fm(x)) for all x{0,1}n.

9. Intuition: We’re given a black box (think: quantum state) f x f(x) 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). This is trivially impossible! But … what if we get 3 black boxes, and are allowed to simulate f=f0 by taking the point-wise MAJORITY of their outputs?

10. “Lifting” the Lemma to Quantumland

11. Quantum Karp-Lipton Theorem:An Unexpected Application of Our BQP/qpoly=YQP/poly Theorem Karp-Lipton 1982: If NPP/poly, then coNPNP = NPNP. Our quantum analogue: If NPBQP/qpoly, then coNPNPQMAPromiseQMA. 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.

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