Quantum Computing and the Limits of the Efficiently Computable

Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson MIT

GOLDBACH CONJECTURE: TRUE NEXT QUESTION Things we never see… Warp drive Übercomputer Perpetuum mobile The (seeming) impossibility of the first two machines reflects fundamental principles of physics—Special Relativity and the Second Law respectively So what about the third one?

Some would say Mr. T & friends already answered this question in the 1930s But what about finding proofs of bounded length? Can that be done in a way that avoids exhaustive search?

P=NP? This sounds like (literally) a $1,000,000 question: If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude.—Gödel to von Neumann, 1956

However, an important presupposition underlying P vs. NP is the... Extended Church-Turing Thesis “Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory” So how sure are we of this thesis? Have there been serious challenges to it?

Old proposal: Dip two glass plates with pegs between them into soapy water. Let the soap bubbles form a minimum Steiner tree connecting the pegs—thereby solving a known NP-hard problem “instantaneously”

Ah, but what about quantum computing?(you knew it was coming) Quantum mechanics: “Probability theory with minus signs” (Nature seems to prefer it that way)

Quantum Computing A quantum state of n qubits takes 2n complex numbers to describe: Chemists and physicists knew that for decades, as a practical problem! In the 1980s, Feynman, Deutsch, and others had the amazing idea of building a new type of computer that could overcome the problem, by itself exploiting the exponentiality inherent in QM Actually building a QC: Damn hard, because of decoherence. (But seems possible in principle!)

Popularizers Beware:A quantum computer is NOT like a massively-parallel classical computer! Exponentially-many basis states, but you only get to observe one of them Any hope for a speedup rides on the magic of interference

Interesting BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by a quantum computer, defined by Bernstein and Vazirani in 1993 Shor 1994: Factoring integers is in BQP NP-complete NP Factoring BQP P

But factoring is not believed to be NP-complete! And today, we don’t believe BQP contains all of NP(though not surprisingly, we can’t prove that it doesn’t) Bennett et al. 1997: “Quantum magic” won’t be enough If you throw away the problem structure, and just consider an abstract “landscape” of 2n possible solutions, then even a quantum computer needs ~2n/2 steps to find the correct one (That bound is actually achievable, using Grover’s algorithm!) So, is there any quantum algorithm for NP-complete problems that would exploit their structure?

Quantum Adiabatic Algorithm(Farhi et al. 2000) Hi Hf Hamiltonian with easily-prepared ground state Ground state encodes solution to NP-complete problem Problem: “Eigenvalue gap” can be exponentially small

Nonlinear variants of the Schrödinger Equation Abrams & Lloyd 1998: If quantum mechanics were nonlinear, one could exploit that to solve NP-complete problems in polynomial time 1 solution to NP-complete problem No solutions

Relativity Computer DONE

Zeno’s Computer STEP 1 STEP 2 Time (seconds) STEP 3 STEP 4 STEP 5

Here’s a polynomial-time algorithm to solve NP-complete problems (only drawback is that it requires time travel): Read an integer x{0,…,2n-1} from the future If x encodes a valid solution, then output x Otherwise, output (x+1) mod 2n Closed Timelike Curves (CTCs) If valid solutions exist, then the only fixed-points of the above program input and output them Building on work of Deutsch, [A.-Watrous 2008] defined a formal model of CTC computation, and showed that in both the classical and quantum cases, it has exactly the power of PSPACE (believed to be even larger than NP)

“The No-SuperSearch Postulate”There is no physical means to solve NP-complete problems in polynomial time. Includes PNP as a special case, but is stronger No longer a purely mathematical conjecture, but also a claim about the laws of physics If true, would “explain” why adiabatic systems have small spectral gaps, the Schrödinger equation is linear, CTCs don’t exist...

Question: What exactly does it mean to “solve” an NP-complete problem? Example: It’s been known for decades that, if you send n identical photons through a network of beamsplitters, the amplitude for the photons to reach some final state is given by the permanent of an nn matrix of complex numbers But the permanent is #P-complete (believed even harder than NP-complete)! So how can Nature do such a thing? Resolution: Amplitudes aren’t directly observable, and require exponentially-many probabilistic trials to estimate

Lesson: If you can’t observe the answer, it doesn’t count! Recently, Alex Arkhipov and I gave evidence that even the observed output distribution of such a linear-optical network would be hard to simulate on a classical computer—but the argument was necessarily subtler Last year, groups in Brisbane, Oxford, Rome, and Vienna reported the first 3-photon BosonSampling experiments, confirming that the amplitudes were given by 3x3 permanents # of experiments > # of photons!

Obvious Challenges for Scaling Up: • Reliable single-photon sources • Minimizing losses • Getting high probability of n-photon coincidence • Goal (in our view): Scale to 10-30 photons • Don’t want to scale much beyond that—both because • you probably can’t without fault-tolerance, and • a classical computer probably couldn’t even verify the results! Theoretical Challenge: Argue that, even with photon losses and messier initial states, you’re still solving a classically-intractable sampling problem

Suppose we believe certain computational problems (e.g., NP-complete ones) are intractable in the physical world. Does that belief do any nontrivial work for physics? Until this January, I thought the answer was yes—but all my examples involved ruling out possibilities (like CTCs) that most physicists consider crazy anyway! But recently, Harlow and Hayden made a striking connection between computational intractability and the black-hole firewall paradox [AMPS 2012] What is this firewall paradox?

Firewalls: Black Hole Information Problem Redux R = Faraway Hawking Radiation H = Just-Emitted Hawking Radiation B = Interior of “Old” Black Hole Near-maximal entanglement Also near-maximal entanglement Violates “monogamy of entanglement”! The same qubit can’t be maximally entangled with 2 things

Harlow-Hayden 2013 (arXiv:1301.4504): Under plausible assumptions about black-hole physics, for Alice to decode the early Hawking radiation R and “see” that it’s entangled with H, she’d need the ability to find “collisions” in a function of the form f:{0,1}n{0,1}n-1 Moreover, I proved in 2002 that, for a “generic” f, the above problem takes exponential time even for a QC! Complexity theory to the rescue of quantum field theory?? Recently I improved the HH argument, to show that Alice’s decoding task is at least as hard as inverting one-way functions (“almost” as hard as NP-complete) “Physical meaning” of these results? Surely a contradiction in physics isn’t OK just because it takes exponential time to find?

Quantum Mechanics Computational Complexity 1990s: Shor & Grover Conclusions Quantum Mechanics Computational Complexity Today: Many other exciting connections between these areas are currently being explored! (Condensed matter, quantum chemistry…) NP Hardness Assumption: Candidate for a robust, fruitful, falsifiable principle bridging complexity and physics?

Quantum Computing and the Limits of the Efficiently Computable