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Home , Boson sampling, BQP, Low (complexity), PR (complexity), QIP (complexity)

On the complexity and verification of Random Circuit Sampling Chinmay Nirkhe [email protected] http://cs.berkeley.edu/~nirkhe Joint work with

Adam Bouland Bill Fefferman Umesh Vazirani UC Berkeley QuICS (U. Maryland/NIST) UC Berkeley

On the Complexity and Verification of Random Circuit Sampling A. Bouland, B. Fefferman, C. Nirkhe, U. Vazirani [Nature Physics 2018] [arXiv:1803.04402] [ITCS 2019] [QIP 2019]

Chinmay Nirkhe (UC Berkeley) 2 Complexity and verification of Random Circuit Sampling Chinmay Nirkhe (UC Berkeley) 3 Complexity and verification of Random Circuit Sampling Chinmay Nirkhe (UC Berkeley) 4 Complexity and verification of Random Circuit Sampling Chinmay Nirkhe (UC Berkeley) 5 Complexity and verification of Random Circuit Sampling Why is there so much hype?

Chinmay Nirkhe (UC Berkeley) 6 Complexity and verification of Random Circuit Sampling The extended Church-Turing thesis

Any ”reasonable” method of computation can be efficiently simulated on a standard model (i.e. , uniform circuits, etc.)

Chinmay Nirkhe (UC Berkeley) 7 Complexity and verification of Random Circuit Sampling The extended Church-Turing thesis

Any ”reasonable”Quantum method of computation can be efficiently simulatedComputing! on a standard model (i.e. Turing machine, uniform circuits, etc.)

∃ 풪 s.t. BPP풪 ≠ BQP풪 [Simon93, Bernstein-Vazirani93]

Chinmay Nirkhe (UC Berkeley) 8 Complexity and verification of Random Circuit Sampling The extended Church-Turing thesis

∃ 풪 s.t. BPP풪 ≠ BQP풪 [Simon93, Bernstein-Vazirani93]

FACTORING ∈ BQP [Shor94]

BQP = the set of languages decidable by a polynomial time quantum

Chinmay Nirkhe (UC Berkeley) 9 Complexity and verification of Random Circuit Sampling The extended Church-Turing thesis

∃ 풪 s.t. BPP풪 ≠ BQP풪 [Simon93, Bernstein-Vazirani93]

FACTORINGSo there is∈ theoreticalBQP [Shor evidence,94] but is there anything tangible? BQP = the set of languages decidable by a polynomial time

Chinmay Nirkhe (UC Berkeley) 10 Complexity and verification of Random Circuit Sampling Experimental progress

72- Bristlecone chip

Chinmay Nirkhe (UC Berkeley) 11 Complexity and verification of Random Circuit Sampling Experimental progress

NISQ Era [Preskill18]

72-qubit Bristlecone chip

Chinmay Nirkhe (UC Berkeley) 12 Complexity and verification of Random Circuit Sampling Oracle Separation

Quantum Experimental Progress Complexity Theory

Quantum Supremacy

Chinmay Nirkhe (UC Berkeley) 13 Complexity and verification of Random Circuit Sampling proposal

A practical demonstration of a quantum computation which is

1. Experimentally feasible 2. Has theoretical evidence of hardness 3. Verifiable

Chinmay Nirkhe (UC Berkeley) 14 Complexity and verification of Random Circuit Sampling Quantum supremacy proposal A practical demonstration of a quantum computation which is

1. Experimentally feasible 2. Has“an theoretical experimental evidence violation of the extended of hardness 3. VerifiableChurch-Turing thesis” – U. Vazirani

Chinmay Nirkhe (UC Berkeley) 15 Complexity and verification of Random Circuit Sampling Why factoring is not the right proposal

The speedups come from carefully Which is hard to generate engineered interference patterns with on the currently available large amounts of constructive and noisy intermediate scale destructive interference quantum devices

“Proving a quantum system’s 7푛 1 computational power by having it This behavior islim far1 +from= “typical”.푒푒푒푒푒푒푒 푛→∞ 푛 factor integers is a bit like proving a It’s hard to make this happen in dolphin’s intelligence by teaching it the lab!! to solve arithmetic problems” [Aaronson-Arkhipov11]

Chinmay Nirkhe (UC Berkeley) 16 Complexity and verification of Random Circuit Sampling Complexity theory Experimentally inspired supremacy inspired supremacy proposals proposals

Problems for which no efficient classical Problems which we can experimentally test algorithms exist (perhaps under complexity- in the near future (~10 years) theoretic conjectures)

Example: [Aaronson11] Example: Random Circuit Sampling [BIS+16]

Proves efficient classical algorithms cannot exist Near-term experimentally feasible due to high-quality unless PH-collapses superconducting

Chinmay Nirkhe (UC Berkeley) 17 Complexity and verification of Random Circuit Sampling A Quantum Supremacy Proposal

Random Circuit Sampling

Given the description of a 퐶, sample from the output distribution of the quantum circuit.

Chinmay Nirkhe (UC Berkeley) 18 Complexity and verification of Random Circuit Sampling Part 0: What is ?

Chinmay Nirkhe (UC Berkeley) 19 Complexity and verification of Random Circuit Sampling What is quantum computing?

It’s computing (really, information processing) based on the principles of quantum mechanics rather than classical physics.

Quantum mechanics is a description of nature • Formulated to explain the behavior of subatomic particles. • QM has been spectacularly successful in explaining microscopic physical phenomena.

Chinmay Nirkhe (UC Berkeley) 20 Complexity and verification of Random Circuit Sampling What is quantum computing? Quantum computers run in superposition. It’s like running probabilistically except there can be negative (complex) probabilities.

Remember physics? =

Chinmay Nirkhe (UC Berkeley) 21 Complexity and verification of Random Circuit Sampling What is quantum computing? The state of a deterministic computation is a binary string 푥 ∈ {0,1}푚 The state of a randomized computation is a

푝푥 푥∈ 0,1 푚 σ푥 푝푥 = 1; 푝푥 ≥ 0

The state of a quantum computation is a superposition

2 ෍ 훼푥|푥⟩ ෍ 훼푥 = 1; 훼푥 ∈ ℂ 푥 푥

Chinmay Nirkhe (UC Berkeley) 22 Complexity and verification of Random Circuit Sampling What is quantum computing?

Quantum computers are realized by measurement. Classical numbers which we can read.

푝푥 푥∈ 0,1 푛 ෍ 훼푥|푥⟩ 푝 = 훼 2 푥∈ 0,1 푛 푥 푥

Chinmay Nirkhe (UC Berkeley) 23 Complexity and verification of Random Circuit Sampling What is quantum computing?

At a high level, quantum computing gives us some of the power of parallel computation without multiple processors.

Some problems have good quantum algorithms, while we believe that for some quantum offers no improvement.

Chinmay Nirkhe (UC Berkeley) 24 Complexity and verification of Random Circuit Sampling Part 1: Classical hardness of Random Circuit Sampling

Chinmay Nirkhe (UC Berkeley) 25 Complexity and verification of Random Circuit Sampling Fix an architecture over quantum circuits

Given a circuit from the architecture, sample from its output probability distribution

Chinmay Nirkhe (UC Berkeley) 26 Complexity and verification of Random Circuit Sampling Goal: show sampling from the exact output distribution of a quantum circuit is #P-hard

Trick: Since proving #P-hardness, by Toda’s Theorem can use PH reductions instead of just P reductions

Chinmay Nirkhe (UC Berkeley) 27 Complexity and verification of Random Circuit Sampling Recall the …

#P

PH ⊆ P#P [Toda91]

Chinmay Nirkhe (UC Berkeley) 28 Complexity and verification of Random Circuit Sampling Goal: show sampling from the exact output distribution of a quantum circuit is #P-hard

Trick: Since proving #P-hardness, by Toda’s Theorem can use PH reductions instead of just P reductions

Chinmay Nirkhe (UC Berkeley) 29 Complexity and verification of Random Circuit Sampling Theorem: Calculating Pr(퐶 outputs 0) is #P-hard.

0 0

0

0

Many proofs: reduce permanents, #CircuitSAT, etc.

Chinmay Nirkhe (UC Berkeley) 30 Complexity and verification of Random Circuit Sampling Stockmeyer’s Theorem85

푥 Proof Sketch:

NP (1) Sufficient to estimate Pr(푥) to multiplicative BPP factor and then amplify. Sampler (2) BPPNP access to a Sampler let’s you

Pr(푥) estimate the # of random 푥 ~ 풟 풟 inputs to the Sampler which output 푥 via random hashing.

Chinmay Nirkhe (UC Berkeley) 31 Complexity and verification of Random Circuit Sampling Exact classical sampling from quantum circuits in P would give us:

Contradicts the non-collapse of the PH:

Toda’s Theorem Pf: Estimating output probabilities is #P-hard. Apply BPPNP reduction due to Stockmeyer’s Thm85 to get sampling is also #P-hard.

Chinmay Nirkhe (UC Berkeley) 32 Complexity and verification of Random Circuit Sampling Therefore, exact quantum sampling is #P-hard under BPPNP-reductions

Chinmay Nirkhe (UC Berkeley) 33 Complexity and verification of Random Circuit Sampling But… No quantum device would exactly sample from the output distribution due to noise!

So in order to make this argument, we have to show that no classical sampler can even approximately sample from the same distribution!

Chinmay Nirkhe (UC Berkeley) 34 Complexity and verification of Random Circuit Sampling By construction, the “hardness” of our circuit was in Pr(0).

If an adversary knew that, they could generate an approximate sampler that never outputted 0.

Thereby, short-circuiting the hardness proof.

Chinmay Nirkhe (UC Berkeley) 35 Complexity and verification of Random Circuit Sampling We want to embed the hardness across the outputs of the probability distribution. Also known as, computing output probabilities should be average-case #P- We want a robustness condition: Being hard. able to compute most probabilities should be #P-hard.

Chinmay Nirkhe (UC Berkeley) 36 Complexity and verification of Random Circuit Sampling Hiding

1 X 0

1 X 0 C C 0 I 0

1 X 0

C1101

Chinmay Nirkhe (UC Berkeley) 37 Complexity and verification of Random Circuit Sampling We want a robustness condition: Being able to compute most probabilities should be #P-hard.

We want a robustness condition: Being able to

compute Pr0(퐶) for most circuits 퐶 should be #푃- hard.

Pr0(퐶) = prob. 퐶 outputs 0

Chinmay Nirkhe (UC Berkeley) 38 Complexity and verification of Random Circuit Sampling What known problem has such a property?

Theorem [Lipton91,GLR+91]: The following is #P-hard: For sufficiently large 푞,

given uniformly random 푛 × 푛 matrix 푀 over 픽푞, output perm(푀) with probability > ¾ + 1/poly(푛)

Chinmay Nirkhe (UC Berkeley) 39 Complexity and verification of Random Circuit Sampling is avg-case hard perm(퐴) is a degree 푛 polynomial in the matrix entries of 퐴

Choose 푅 a . Let 푀(푡) = 퐴 + 푅푡. 푀(0) = 퐴 and 푀(푡) for 푡 ≠ 0 is uniformly random. perm(푀(푡)) is a degree 푛 polynomial in 푡

Choose random 푡1, … , 푡푛+1, calculate perm(푀(푡푖)) Interpolate the polynomial perm(푀(푡)). Output perm(푀(0))

Chinmay Nirkhe (UC Berkeley) 40 Complexity and verification of Random Circuit Sampling Permanent is avg-case hard

Assume we can calculate perm(푅) for random 푅 with probability > 1 − 1 /(3푛 + 3). By union bound, we calculate perm(퐴) with probability 2/3. Since permanent is worst-case #P-hard [Valiant79], This proves statement for probability 1 – 1/(3푛 + 3).

Better interpolation techniques bring the probability down to ¾ + 1/poly(푛).

Chinmay Nirkhe (UC Berkeley) 41 Complexity and verification of Random Circuit Sampling Goal: find a similar polynomial structure in the problem of Random Circuit Sampling

Chinmay Nirkhe (UC Berkeley) 42 Complexity and verification of Random Circuit Sampling High-level idea

Feynman Path Integral:

Quantum analog of space-efficient brute-force evaluation of a circuit

Chinmay Nirkhe (UC Berkeley) 43 Complexity and verification of Random Circuit Sampling Feynman Path Integral

Then Pr0(퐶) is a -degree polynomial in the gate entries. We want to apply a similar interpolation technique as permanents.

Chinmay Nirkhe (UC Berkeley) 44 Complexity and verification of Random Circuit Sampling Idea #1

Consider the circuit 퐶(푡) formed by changing each gate 퐶푖 to 퐶푖 + 푡퐻푖 for random gate 퐻푖.

Just like permanent!

But, 퐶푖 + 푡퐻푖 isn’t a quantum gate!

Chinmay Nirkhe (UC Berkeley) 45 Complexity and verification of Random Circuit Sampling Idea #2

Consider the circuit 퐶(휃) formed by changing each gate

1. 퐶(1) = 퐶 2. For small 휃, circuit looks 휃-close to random! 3. Not a low-degree polynomial in 휃

Chinmay Nirkhe (UC Berkeley) 46 Complexity and verification of Random Circuit Sampling Idea #3 Taylor Series!

1. 퐶(1) ≈ 퐶 2. For small 휃, circuit looks 휃-close to random! 3. A low-degree polynomial in 휃 4. For more complicated technical reasons, this is a necessary, but not sufficient, proof of average-case hardness.

Chinmay Nirkhe (UC Berkeley) 47 Complexity and verification of Random Circuit Sampling Theorem: On average, it is hard to exactly sample from the output of random circuits.

This puts Random Circuit Sampling on par with the best known supremacy proposals.

Chinmay Nirkhe (UC Berkeley) 48 Complexity and verification of Random Circuit Sampling Part 2: Verification of Random Circuit Sampling

Chinmay Nirkhe (UC Berkeley) 49 Complexity and verification of Random Circuit Sampling How do we know if our quantum computer is doing what it says its doing?

Chinmay Nirkhe (UC Berkeley) 50 Complexity and verification of Random Circuit Sampling phdcomics.com & Caltech IQIM

Chinmay Nirkhe (UC Berkeley) 51 Complexity and verification of Random Circuit Sampling phdcomics.com & Caltech IQIM

Chinmay Nirkhe (UC Berkeley) 52 Complexity and verification of Random Circuit Sampling Sweet spot in verification

Compromise: OK with exponential post Toolbox processing time on supercomputer to compute “a few” ideal output probabilities for “intermediate” size quantum computers (푛 = 50 qubits) Noisy 50 qubit quantum computer Constraint: can only take a small (poly(푛)) number of samples from the quantum device Super-computers capable of 260 size Challenge: Complexity arguments require computations closeness in total variation distance. But we can’t hope to unconditionally verify this with few samples from the device.

Chinmay Nirkhe (UC Berkeley) 53 Complexity and verification of Random Circuit Sampling Candidate test: Cross-Entropy [Boixo+16]

1 1 CE 푝dev, 푝id = ෍ 푝dev 푥 log = 피푝dev log 푝id(푥) 푝id 푥

Can be approximated in a few samples • Sample 푥1, … , 푥푘from quantum device • Use exponential time to calculate 푝id(푥푖) • Estimate CE

• If score is close enough to expected CEideal, then accept.

Chinmay Nirkhe (UC Berkeley) 54 Complexity and verification of Random Circuit Sampling Candidate test: Cross-Entropy [Boixo+16]

This is a one-dimensional projection of high-dimensional information

Does not verify closeness in total-variation distance

Theorem: If cross-entropy is close to ideal and 퐻 푝dev ≥ 퐻(푝id) then the output distribution is close to ideal in total variation distance

Chinmay Nirkhe (UC Berkeley) 55 Complexity and verification of Random Circuit Sampling Candidate test: Binned output generation

Consider dividing the [0,1] interval into poly(푛) bins

Observe 푘 samples 푥1, … , 푥푘 and Thanks!calculate ideal probabilities for each sample on supercomputer

Plot and make sure correct number of points in each bin

Chinmay Nirkhe (UC Berkeley) 56 Complexity and verification of Random Circuit Sampling