Quantum Advantage with Shallow Circuits

Quantum advantage with shallow circuits

Robert König

joint work with Sergey Bravyi and David Gosset (IBM)

QIP 2018

arXiv:1704.00690 Are quantum computers more powerful than classical ones?

The development of a scientific field – A familiar cycle of seasons …but will the progress flourish into a quantum summer?

Justified excitement, or overblown promise?

[… ] how realistic are such claims about the future of quantum computing?

[A. Abbott and C. Calude, Limits of Quantum Computing: A Sceptic’s View http://www.quantumforquants.org/quantum-computing/limits-of-quantum-computing/ ] “three good reasons for thinking that quantum computers have capabilities surpassing what classical computers can do”

(1) Quantum algorithms for classically intractable problems. First, we know of problems that are believed to be hard for classical computers, but for which quantum algorithms have been discovered that could solve these problems easily. The best known example is the problem of finding the prime factors of a large composite integer [1]. We believe factoring is hard because many smart people have tried for many decades to find better factoring algorithms and haven’t succeeded. Perhaps a fast classical factoring algorithm will be discovered in the future, but that would be a big surprise.

(2) Complexity theory arguments. The theoretical computer scientists have provided arguments, based on complexity theory, showing (under reasonable assumptions) that quantum states which are easy to prepare with a quantum computer have superclassical properties; specifically, if we measure all the qubits in such a state we are sampling from a correlated probability distribution that can’t be sampled from by any efficient classical means [2, 3].

(3) No known classical algorithm can simulate a quantum computer. But perhaps the most persuasive argument we have that quantum computing is powerful is simply that we don’t know how to simulate a quantum computer using a digital computer; that remains true even after many decades of effort by physicists to find better ways to simulate quantum systems. [J. Preskill, arXiv:1801.00862 (based on a keynote address at Quantum Computing for Business)] Bernstein-Vazirani problem (1993) Problem: Find 푧 ∈ {0,1}푛 using few queries to an oracle: Linear Boolean function ℓ 푇 |푥⟩ −1 푧 푥|푥⟩ parameterized by a “secret” bit 푈ℓ string 푧

• The following quantum circuit • In contrast, any classical algorithm computes z using only 1 query to the needs 풏 queries to a classical quantum oracle: oracle computing ℓ to determine z.

[Bernstein and Vazirani, Quantum 푛 ⊗푛 ⊗푛 푛 |0 ⟩ 퐻 퐻 푧 ∈ {0,1} complexity theory, SIAM Journal on 푈ℓ Computing, 26(5):1411-1473, 1997]

This problem shows a separation between classical and quantum algorithms in terms of query complexity. Potential concerns about query complexity separations

“Where’s my black-box?” 푇 |푥⟩ −1 푧 푥|푥⟩ 푈ℓ

“Black-box problems, where one is required to compute a function or property of a classical input by querying a quantum box, are easier to prove speedups for because of their extra formal structure. But this also means that they often have a somewhat artificial flavour and their practical relevance is questionable.”

“One well-known such algorithm is Grover’s algorithm, [….] The cost of constructing the quantum database could negate any advantage of the algorithm, and in many classical scenarios one could do much better by simply creating (and maintaining) an ordered database.” [A. Abbott and C. Calude, Limits of Quantum Computing: A Sceptic’s View http://www.quantumforquants.org/quantum-computing/limits-of-quantum-computing/ ] Our result

A provable, non-oracular quantum speedup,

attainable by a

constant-depth

geometrically local (in a 2D),

circuit. This talk: constant-depth (quantum) circuits

A depth-풅 quantum circuit consists of 푑 time steps. Each time step contains one- and two-qubit gates acting on disjoint qubits.

푘-bit input |푥⟩ … 푛-bit output 푧

Example: depth-5 circuit

output probability distribution for a given input 푥

Family of circuits 푈푛 푛 with depth 푶 ퟏ . Constant-depth or “Shallow”

Fixed set of gates independent of 푛. Motivation for considering constant-depth quantum circuits:

The Noisy Intermediate-Scale Quantum (NISQ) Technology Era

[J. Preskill, Quantum Computing in the NISQ era and beyond, arXiv:1801.00862 ] Circuit depth in the Noisy Intermediate-Scale Quantum Technology Era Noise sets a limit on the maximum size of a computation without error correction. 푛 = number of qubits (width) Rough estimate: 푛푑 ≪ 1/휖 푑 = circuit depth 휖 = error rate

1 … 푛 fault fault

Deep circuits → few qubits → efficient classical simulation. Shallow circuits → many qubits → potential for a quantum advantage. Shallow circuits and their potential 푑 ≤ 2 푂(1) 푂(log 푛) 푝표푙푦(푛)

approximate Shor’s algorithm optimization efficient Clifford circuits BQP classical variational quantum good variational complete simulation eigensolver states (MERA) quantum toy models of supremacy ? black holes ? Farhi, Goldstone, Gutmann 2014 Cleve, Watrous 2000 Terhal, Peruzzo et al 2014 DiVincenzo 2002 Vidal 2007 Boixo et al 2016 Moore, Nilsson 2001 Bermejo-Vega et al 2017 Brown, Fawzi 2013 Gao, Wang, Duan 2017 Bremner, Montanaro, Shepherd 2016 Constant-depth quantum circuits versus classical circuits 푂(1) 푝표푙푦(푛) Quantum circuits (depth) Can constant-depth quantum circuits solve a computational ? problem that polynomial-time classical computations Classical computation (time) cannot?

Too difficult question… No hope for unconditional proof.

Positive answer would imply P≠PSPACE. Constant-depth quantum versus classical circuits

푂(1) 푝표푙푦(푛) Quantum circuits (depth) Can constant-depth quantum circuits solve a computational problem that constant-depth classical circuits cannot? Classical circuits (depth)

This talk: The answer is YES. Classical circuits

A classical gate computes a Boolean function 푓: {0,1}푘 → {0,1}

푥1 푓(푥) Number of times the output is used is called Number of input bits k 푥2 푓(푥) is called the fan-in ⋮ f the fan-out 푥푘

We consider circuits composed of bounded fan-in gates, i.e., 푘 = 푂(1).

We do not restrict the fan-out. Constant-depth classical circuits

A depth-푑 classical circuit consists of 푑 layers (time steps) of gates.

OR AND

AND OR

Time step 1 Time step 2 Constant-depth classical circuits

A depth-푑 classical circuit consists of 푑 layers (time steps) of gates.

OR AND

AND OR

We consider constant-depth circuits composed of bounded fan-in gates.

We also allow the circuit to be probabilistic (random input bits are provided). Can constant-depth quantum circuits solve a computational problem that constant-depth classical circuits cannot?

Input Output

Decision problem Bit-string x 푏푥 ∈ {0,1}

Causality: The marginal distribution of any output bit is determined by 푂 1 input bits. Example: S Simulation by a classical S circuit of size 푂(1) H S Can constant-depth quantum circuits solve a computational problem that constant-depth classical circuits cannot?

Input Output

Decision problem Bit-string x 푧 ∈ {0,1}

Causality: The marginal distribution of any output bit is determined by 푂 1 input bits. S Support of S size H S 푂푧 2푑 = 푂 1 O(1)-local operator Can constant-depth quantum circuits solve a computational problem that constant-depth classical circuits cannot?

Input Output

Decision problem Bit-string x 푏푥 ∈ {0,1}

푛 Relation problem Bit-string x 푧 ∈ 푆푥 ⊆ {0,1} (non-unique)

Example: combinatorial optimization, say 3-SAT

푥 system of equations 푆 set of bit strings 푧 ∈ 푆 a satisfying with binary variables 푥 satisfying all equations 푥 assignment Can constant-depth quantum circuits solve a computational problem that constant-depth classical circuits cannot?

Input Output

Decision problem Bit-string x 푏푥 ∈ {0,1}

푛 Relation problem Bit-string x 푧 ∈ 푆푥 ⊆ {0,1} (non-unique)

A (quantum) circuit solves a relation problem if

for any input 푥 it outputs a valid solution 푧 (with high probability) : Quantum advantage of constant-depth circuits

Quantum circuit Classical needs depth depth 풅 = 푶(ퟏ) 퐝 ≥ 풄퐥퐨퐠 퐧

S OR AND S AND OR H S Our result: We describe a (relation) problem such that

• The problem is solved with certainty (휖 = 0) by a constant-depth quantum circuit (with geometrically local gates in 2D).

• Any probabilistic classical circuit composed of bounded fan-in gates (possibly non-local) which solves the problem with high probability (휖 < 1/8) must have depth increasing logarithmically with input size.

The quantum speedup is unconditional: It is non-oracular and does not rely on complexity-theoretic conjectures.

Note: The problem can be solved in polynomial time classically. The Hidden Linear Function (HLF) Problem The Bernstein-Vazirani speedup is relative to an oracle.

Linear Boolean function parameterized by a “secret” bit string 푧

푇 |푥⟩ −1 푧 푥|푥⟩ 푈ℓ

Where else can we hide a linear function? Binary quadratic forms

Suppose 퐴 is a symmetric binary matrix of size 푛

Nullspace:

Quadratic form:

1 1 0 0 Example: 퐴 = 1 0 1 퐴푥 = 0 (mod 2) Ker 퐴 = 000, 111 0 1 1 0 푇 푞 푥 = 푥 퐴 푥 mod 4 = 푥1 + 푥3 + 2푥1푥2 + 2푥2푥3 (mod 4) Real-valued versus binary quadratic forms

푥 is a real vector: 퐴푥 = 0푛 implies 푥푇퐴푥 = 0.

푥 is binary: the restriction of 푞(푥) onto Ker 퐴 can be non-zero

1 1 0 0 Example: 퐴 = 1 0 1 퐴푥 = 0 (mod 2) Ker 퐴 = 000, 111 0 1 1 0 푇 푞 푥 = 푥 퐴 푥 mod 4 = 푥1 + 푥3 + 2푥1푥2 + 2푥2푥3 (mod 4)

푞 111 = 1 + 1 + 2 + 2 mod 4 = 2 Fact: The restriction of 푞(푥) onto the nullspace of 퐴 is a linear function (up to a factor of 2) Boolean linear function

Proof sketch: for

since Fact: The restriction of 푞(푥) onto the nullspace of 퐴 is a linear function (up to a factor of 2) Boolean linear function

A binary quadratic form hides a Boolean linear function (in a non-oracular way) Fact: The restriction of 푞(푥) onto the nullspace of 퐴 is a linear function (up to a factor of 2) “secret” bit string

Hidden Linear Function (HLF) problem Input: binary symmetric matrix 퐴. Output: bitstring 푧 such that 푞 푥 = 2푧푇푥 (mod 4) for all 푥 ∈ Ker(퐴) Hidden Linear Function (HLF) problem Input: binary symmetric matrix 퐴. Output: bitstring 푧 such that 푞 푥 = 2푧푇푥 (mod 4) for all 푥 ∈ Ker(퐴)

• This can be viewed as a non-oracular variant of the Bernstein-Vazirani problem.

• The solution is non-unique:

if 푧 is a solution and 푦 ∈ Ker 퐴 ⊥ then 푧 ⊕ 푦 is a solution

• The HLF problem can be solved classically in time 푂 푛3

1) Compute a basis 푏1, … , 푏푘 of the nullspace Ker(퐴) 2) Solve a linear system 2 푧푇푏푖 = 푞(푏푖), 푖 = 1, … , 푘 The 2D HLF: quadratic forms on a square grid

Consider a square grid of size 푛 × 푛

Variables 푥1, … , 푥푛 live at sites

퐴푖,푗 = 0 unless (푖, 푗) are nearest neighbors

푖 푗 퐴푖.푗 = 1

푛 sites 푖 퐴푖.푖 = 1

The 2D HLF problem is the set of instances where the (off-diagonal part of the) matrix A is the adjacency matrix of a subgraph of the 푛 × 푛 grid graph. Remainder of the talk

• The hidden linear function (HLF) problem

• A quantum algorithm for the 2D HLF Problem (constant-depth circuit)

• Proof of hardness for constant-depth classical circuits Solving the 2D HLF by a constant-depth quantum circuit

퐴

|0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}푛

Apply CZ gates on Apply phase gates 푆푖 on edges with 퐴푖,푗 = 1 vertices with 퐴푖,푖 = 1

Gate set: Clifford gates H, S, CZ with one (classical) control bit. Solving the 2D HLF by a constant-depth quantum circuit

퐴

|0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}푛

Apply CZ gates on Apply phase gates 푆푖 on edges with 퐴푖,푗 = 1 vertices with 퐴푖,푖 = 1

Fact 1: The string of measurement outcomes 푧 is a solution to the 2D HLF Problem. (The distribution 푃(푧|퐴) is uniform on the set of all solutions.)

Fact 2: The circuit can be implemented in constant depth. (with nearest neighbor gates in 2D) Solving the 2D HLF by a constant-depth quantum circuit

퐴

푛 |0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}

Apply CZ gates on Apply phase gates 푆푖 on edges with 퐴푖,푗 = 1 vertices with 퐴푖,푖 = 1

Fact 1: The string of measurement outcomes 푧 is a solution to the 2D HLF Problem.

similar to IQP circuits Bremner, Montanaro, Shepherd 2016 Solving the 2D HLF by a constant-depth quantum circuit

퐴

푛 |0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}

Apply CZ gates on Apply phase gates 푆푖 on edges with 퐴푖,푗 = 1 vertices with 퐴푖,푖 = 1

Fact 1: The string of measurement outcomes 푧 is a solution to the 2D HLF Problem.

Relationship to IQP circuits:

The unitary is diagonal and satisfies for all

This (explicitly realized) unitary takes the place of an oracle in the Bernstein-Vazirani algorithm. Solving the 2D HLF by a constant-depth quantum circuit

퐴

푛 |0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}

Apply CZ gates on Apply phase gates 푆푖 on edges with 퐴푖,푗 = 1 vertices with 퐴푖,푖 = 1

Fact 1: The string of measurement outcomes 푧 is a solution to the 2D HLF Problem.

similar to IQP circuits Bremner, Montanaro,Shepherd 2016 Solving the 2D HLF by a constant-depth quantum circuit

퐴

|0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}푛

Apply CZ gates on Apply phase gates 푆푖 on edges with 퐴푖,푗 = 1 vertices with 퐴푖,푖 = 1

Fact 1: The string of measurement outcomes 푧 is a solution to the 2D HLF Problem.

use the fact that 푞(푥) is a quadratic form Solving the 2D HLF by a constant-depth quantum circuit

퐴

|0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}푛

Apply CZ gates on Apply phase gates 푆푖 on edges with 퐴푖,푗 = 1 vertices with 퐴푖,푖 = 1

Fact 1: The string of measurement outcomes 푧 is a solution to the 2D HLF Problem.

use the fact that the restriction of 푞 푥 to Ker(퐴) is linear Solving the 2D HLF by a constant-depth quantum circuit

퐴

|0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}푛

Apply CZ gates on Apply phase gates 푆푖 on edges with 퐴푖,푗 = 1 vertices with 퐴푖,푖 = 1

Fact 1: The string of measurement outcomes 푧 is a solution to the 2D HLF Problem.

similar to Bernstein-Vazirani Solving the 2D HLF by a constant-depth quantum circuit

퐴

|0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}푛

1 time-step 1 time-step 1 time-step

Four layers of CCZ gates. (even/odd vertical/horizontal edges) Decompose CCZ gates into 1- and 2-qubit gates.

Fact 2: The circuit can be implemented in constant depth (with nearest neighbor gates in 2D) A constant-depth quantum circuit for 2D HLF

Place a qubit at each vertex in |0⟩

Place input bits on vertices and edges:

: Edge with 퐴푖,푗= 1

: Vertex with 퐴푖,푖 = 1 A constant-depth quantum circuit for 2D HLF

apply H to every qubit 퐻 퐻 퐻 퐻 퐻

퐻 퐻 퐻 퐻 퐻 : Edge with 퐴푖,푗= 1

: Vertex with 퐴푖,푖 = 1 퐻 퐻 퐻 퐻 퐻

퐻 퐻 퐻 퐻 퐻

퐻 퐻 퐻 퐻 퐻 Only requires classically controlled Clifford gates between nearest neighbor qubits on a 2D grid. A constant-depth quantum circuit for 2D HLF

apply a CZ to every pair (푖, 푗) of qubits 퐶푍 with 퐴푖,푗= 1

: Edge with 퐴푖,푗= 1

: Vertex with 퐴푖,푖 = 1 퐶푍

퐶푍 Only requires classically controlled Clifford gates between nearest neighbor qubits on a 2D grid. A constant-depth quantum circuit for 2D HLF

apply a CZ to every pair (푖, 푗) of qubits with 퐴푖,푗= 1

퐶푍 : Edge with 퐴푖,푗= 1

: Vertex with 퐴푖,푖 = 1

Only requires classically controlled Clifford gates between nearest neighbor qubits on a 2D grid. A constant-depth quantum circuit for 2D HLF

apply a CZ to every pair (푖, 푗) of qubits with 퐴푖,푗= 1

: Edge with 퐴푖,푗= 1 퐶푍 퐶푍 : Vertex with 퐴푖,푖 = 1

Only requires classically controlled Clifford gates between nearest neighbor qubits on a 2D grid. A constant-depth quantum circuit for 2D HLF

apply a CZ to every pair (푖, 푗) of qubits with 퐴푖,푗= 1 퐶푍

: Edge with 퐴푖,푗= 1

: Vertex with 퐴푖,푖 = 1

퐶푍

Only requires classically controlled Clifford gates between nearest neighbor qubits on a 2D grid. A constant-depth quantum circuit for 2D HLF

apply S to every qubit 푖 with 퐴푖,푖= 1 푆

: Edge with 퐴푖,푗= 1

: Vertex with 퐴푖,푖 = 1 푆 푆

푆 푆

Only requires classically controlled Clifford gates between nearest neighbor qubits on a 2D grid. A constant-depth quantum circuit for 2D HLF

apply H to every qubit 퐻 퐻 퐻 퐻 퐻

퐻 퐻 퐻 퐻 퐻 : Edge with 퐴푖,푗= 1

: Vertex with 퐴푖,푖 = 1 퐻 퐻 퐻 퐻 퐻

퐻 퐻 퐻 퐻 퐻

퐻 퐻 퐻 퐻 퐻 Only requires classically controlled Clifford gates between nearest neighbor qubits on a 2D grid. A constant-depth quantum circuit for 2D HLF

measure each qubit in the computational basis

: Edge with 퐴푖,푗= 1

: Vertex with 퐴푖,푖 = 1

Only requires classically controlled Clifford gates between nearest neighbor qubits on a 2D grid. The HLF circuit and graph states

Quantum algorithm solving the HLF :

퐴

|0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}푛

Prepare the graph state Measure qubit 푖 in the X basis if 퐴푖,푖 = 0 for a graph with adjacency Measure qubit 푖 in the Y basis if 퐴푖,푖 = 1 matrix 퐴

M. Hein, J. Eisert, H.J. Briegel: Multi-party entanglement in graph states, Phys. Rev. A 69, 062311 (2004) R. Raussendorf, D.E. Browne, H.J. Briegel: Measurement-based quantum computation on cluster states, Phys. Rev. A 69, 062311 (2004) The HLF circuit and graph states

Quantum algorithm solving the HLF :

퐴

|0푛⟩ 퐻⊗푛 퐶푍(퐴) 푆(퐴) 퐻⊗푛 푧 ∈ {0,1}푛 input: Prepare the graph state Measure qubit 푖 in the X basis if 퐴푖,푖 = 0 for a graph with adjacency Measure qubit 푖 in the Y basis if 퐴푖,푖 = 1 : Ai,j = 1 matrix 퐴 : A푖,푖 = 1

measure X

measure Y Remainder of the talk

• The hidden linear function (HLF) problem

• A quantum algorithm for the 2D HLF Problem (constant-depth circuit)

• Proof of hardness for constant-depth classical circuits Main result: A lower bound on classical circuits Theorem: The following holds for all sufficiently large 푛.

Let 풞푛 be a classical probabilistic circuit where each gate of 풞푛 has fan-in at most 푲. Suppose it solves size-풏 instances of the 2D HLF Problem with probability > ퟕ/ퟖ. Then log(푛) depth 풞 ≥ 푛 16 log(퐾)

Input 퐴 Output 푛 풞푛 푧 ∈ {0,1} ℓ Random bits 푟 ∈ {0,1} Solution with (drawn from any probability > ퟕ/ퟖ joint distribution) For constant fan-in: Circuit must have depth 훀(퐥퐨퐠 퐧 ) Proof idea

Locality in shallow classical circuits Quantum non-locality

John Bell David Mermin OR AND versus

AND OR

Each output bit depends only on Measurement statistics of O(1) input bits. entangled quantum states cannot be reproduced by local hidden variable models. We will show that quantum nonlocality beats

(A) Strictly local classical circuits (local hidden variable models)

(B) Geometrically local classical circuits in 1D

푥1 푥2 OR AND 푧1 input 푥3 output 푥4 AND OR 푧2

푥5 AND 푥6 Locality in classical circuits

푥1 푥2 OR AND 푧1

퐿(푥5) input 푥3 푥4 AND OR 푧2

푥5 AND 푥6

The (forward) lightcone 퐿 푥푘 of an input bit 푥푘 is the set of output bits 풛풊 that are causally connected to 풙풌. Locality in classical circuits

푥1 푥2 OR AND 푧1

푥3 output 푥 푧 퐿(푧1) 4 AND OR 2

푥5 AND 푥6

The (backward) lightcone 퐿 푧푘 of an output bit 푧푘 is the set of input bits 풙풊 that are causally connected to 풛풌. We will show that quantum nonlocality beats

(A) Strictly local classical circuits 퐿(푧 ) = {푥 } 푧 (local hidden variable models) 푘 푘 for any output bit 푘

퐷 (B) Geometrically local 퐿(푥푘) ⊂ 퐵 (푥푘) for any input bit 푥푘 classical circuits in 1D

푑 (C) “Constant-depth local” |퐿(푧푘)| ≤ 퐾 for all output bits 푧푘 classical circuits Strictly local classical circuits

A strictly local circuit has the property that 퐿(푧푘) = {푥푘} for any output bit 푧푘

(We assume here that there is a one-to-one correspondence between input- and output bits.)

푥1 푧1 Note: Every output bit 푧푘 is of the form 푧푘 = 푓푘 푥푘 푥 2 푧2

푥3 푧3 Strictly local classical circuits

A strictly local circuit has the property that 퐿(푧푘) = {푥푘} for any output bit 푧푘

(We assume here that there is a one-to-one correspondence between input- and output bits.)

푥1 푓1 푧1 Note: Every output bit 푧푘 is of the form 푧푘 = 푓푘 푥푘 푥2 푓2 푧2

푥3 푓3 푧3 Strictly local classical circuits

A strictly local circuit has the property that 퐿(푧푘) = {푥푘} for any output bit 푧푘

(We assume here that there is a one-to-one correspondence between input- and output bits.)

푥1 푓1 푧1 Note: If the circuit is probabilistic, 푥2 푓2 푧2 푟 every output bit 푧푘 is of the form 푧푘 = 푓푘 푥푘, 푟 푥3 푓3 푧3 where 푟 is shared randomness

푟 = shared random bit string Circuits and the GHZ relation [Greenburger et al. 1990][Mermin 1990]

GHZ-relation: Compact form

풙 풙 풙 풛 풛 풛 푥1 푧1 ퟏ ퟐ ퟑ ퟏ ퟐ ퟑ ퟎ ퟎ ퟎ 1 푅 푥, 푧 = 1 ퟏ ퟏ ퟎ −1 푥 2 풞 푧2 ퟏ ퟎ ퟏ −1 whenever 3 푥 ⊕ 푥 ⊕ 푥 = 0 ퟎ ퟏ ퟏ −1 1 2 3

푥3 푧3 inputs 푥푖 ∈ 0,1 outputs 푧푖 ∈ {+1, −1}

Suppose 푪 obeys the GHZ relation for any input 푥.

What can be said about its locality ? [Greenburger et al. 1990][Mermin 1990] Strictly local circuits and the GHZ relation

GHZ-relation:

풙 풙 풙 풛 풛 풛 푥1 푧1 ퟏ ퟐ ퟑ ퟏ ퟐ ퟑ Impossible for ퟎ ퟎ ퟎ 1 Local Hidden ퟏ ퟏ ퟎ −1 푥 푧 Variable Models! 2 푟 2 ퟏ ퟎ ퟏ −1 ퟎ ퟏ ퟏ −1

푥3 푧3 inputs 푥푖 ∈ 0,1 outputs 푧푖 ∈ {+1, −1} 푟 = shared random bit string reinterpeted as Corollary: There is no strictly local classical circuit a limitation of strictly local circuits: outputting z satisfying the GHZ relation on all inputs x. Circuits and the GHZ relation

Suppose a classical 푥1 푧1 Lemma: probabilistic circuit satisfies the GHZ relation with probability > 7/8. 푥2 푧2 Then the lightcone 퐿(푥푖) of some input bit 푥푖 contains a distinct output bit 푧푘, that is, 푖 ≠ 푘 푥3 풞3 푧3

Corollary: There is no strictly local classical circuit outputting z satisfying the GHZ relation on all inputs x. Circuits and the GHZ relation

Corollary: There is no strictly local classical circuit outputting z satisfying the GHZ relation on all inputs x. [Greenburger et al. 1990][Mermin 1990] Satisfying the GHZ relation with quantum non-locality

푥 1 푧1 1) Prepare |GHZ⟩

|GHZ⟩ 푥2 푧2 2) Measure each qubit of |퐺퐻푍〉 in either the X basis (if 푥푗 = 0) or the Y basis (if 푥푗 = 1). 푥3 푧3

Outcomes 풛풋 ∈ −ퟏ, +ퟏ 1 퐺퐻푍 = 000 + |111⟩ satisfy the GHZ relation √2 Quantum nonlocality beats strictly local circuits GHZ-relation: 푥 푧 1 1 풙ퟏ 풙ퟐ 풙ퟑ 풛ퟏ풛ퟐ풛ퟑ 푥1 푧1 ퟎ ퟎ ퟎ 1 푥2 푧2 ퟏ ퟏ ퟎ −1 푥2 푧2 |퐺퐻푍〉 ퟏ ퟎ ퟏ −1 풞3 푥3 푧3 ퟎ ퟏ ퟏ −1 푥3 푧3

Lemma: Suppose a classical Lemma: This quantum algorithm probabilistic circuit satisfies the GHZ produces an element z satisfying the relation with probability > 7/8. GHZ relation with probability 1: 1) Prepare |GHZ⟩ Then the lightcone 퐿(푥푖) of some input 2) Measure each qubit of |GHZ⟩ bit 푥푖 contains a distinct output bit 푧푘, in either the X basis (if 푥푗 = 0) 푖 ≠ 푘 that is, or the Y basis (if 푥푗 = 1). We will show that quantum nonlocality beats

(A) Strictly local classical circuits 퐿(푧 ) = {푥 } 푧 (local hidden variable models) 푘 푘 for any output bit 푘

퐷 (B) Geometrically local 퐿(푥푘) ⊂ 퐵 (푥푘) for any input bit 푥푘 classical circuits in 1D

푑 (C) “Constant-depth local” |퐿(푧푘)| ≤ 퐾 for all output bits 푧푘 classical circuits Geometrically local circuits Assumption: Input- and Output bits are associated with vertices of a graph 훿(푥, 푦) ≔ Graph distance between 푥 und 푦 퐵퐷(푥) ≔ 푦 훿(푥, 푦) ≤ 퐷}

퐷 A geometrically 푫 −local circuit satisfies 퐿(푥푘) ⊂ 퐵 (푥푘) for any input bit 푥푘

Example: geometric locality in 1 dimension

푥1 푧1 푥 2 푧2 푥푘 퐷 = 푂(1) 퐿(푥퐿(푥푘) Shallow circuit with nearest-neighbor gates in 1D

depth 푑 = 푂 1 푥푛 푧푛 Quantum nonlocality beats geometrically local circuits J. Barret, C. Caves, B. Eastin, M. Elliot, S. Pironio: Modeling Pauli measurements on graph states with nearest-neighbor classical communication, PRA 75, 012103, 2007.

Simulating the measurement statistics resulting from a graph state cannot be achieved with limited-distance classical communication between nodes

Here we reinterpret this result as a limitation of geometrically local circuits. GHZ generalized: the cycle relation [Barrett et al. 2007] 푛 even. Consider the 푛-cycle and 푤 푢, 푣, 푤 on the even sublattice 푡1 푠3 3 “Input”: 푥 = (푥푢, 푥푣, 푥푤) ∈ {0,1}

푡3 푠1 “Output”: z ∈ {−1,1}|푉|= {−1,1}푛 푟 푢 푟1 3 푣

Define the “cycle relation”: 푅 푥, 푧 = 1 whenever

Fact: Satisfying the cycle relation Fact: The cycle relation is satisfied when (classically) requires communication. (appropriately) measuring the cycle graph state. (Geometrically local) circuits and the cycle relation 푥푖 Lemma: Suppose a classical circuit satisfies the cycle relation 푤 with probability > 7/8 for each input. Then the lightcone 퐿(푥푖) of some input bit 푥푖 , 푖 ∈ {푢, 푣, 푤} contains a distant output bit 푧푘. 푧푘

More precisely: 훿(푥푖, 푧푘) > 퐷, 퐷 = min{ 훿 푢, 푣 , 훿 푣, 푤 , 훿 푤, 푣 }/2 푢 푣

Corollary: There is no geometrically 푫-local (probabilistic) classical circuit giving an output 푧 satisfying the cycle relation R 푥, 푧 = 1 for all inputs 푥 ∈ 0,1 3. (Geometrically local) circuits and the cycle relation 푥푖 Lemma: Suppose a classical circuit satisfies the cycle relation 푤 with probability > 7/8 for each input. Then the lightcone 퐿(푥푖) of some input bit 푥푖 , 푖 ∈ {푢, 푣, 푤} contains a distant output bit 푧푘. 푧푘

More precisely: 훿(푥푖, 푧푘) > 퐷, 퐷 = min{ 훿 푢, 푣 , 훿 푣, 푤 , 훿 푤, 푣 }/2 푢 푣

Proof idea: Assume this is not the case. Then the (relevant part of) 푤 the output of the circuit can be described by four functions

푡1 푠3

푡3 푠1

푟 such that 푢 푟1 3 푣 Satisfying the cycle relation with quantum non-locality [Barrett et al. 2007]

푛 Φ = ෑ 퐶푍 퐻⊗푛|0푛⟩ 1) Prepare the graph state 푛 푗,푗+1 푋/푌 푤 associated with the cyle 푗=1 푋 푋 2) For each qubit 푗 ∈ {푢, 푣, 푤} measure 푋 푋 for any other qubit measure 푢 푋 푋 푣 푋/푌 푋 푋 푋 푋/푌 Call 푧푗 the measurement outcome for qubit 푗.

Fact: This quantum algorithm produces outcomes z satisfying the cycle relation 푅 푥, 푧 = 1

Intuition: the reduced state of 푢푣푤 (after all green qubits are measured) is the GHZ state modulo a Pauli correction that depends on the measurement outcomes. Quantum nonlocality beats geometrically local circuits

푥푖 Cycle-relation 푋/푌 푤 푤 푅 푥, 푧 = 1 whenever 푋 푋 푧푘 푋 푋 푢 푋 푋 푣 푢 푣 푋/푌 푋 푋 푋 푋/푌

Lemma: Suppose a classical circuit Lemma: This quantum algorithm satisfies the cycle relation with produces an element z satisfying the probability > 7/8 for each input. cycle relation with probability 1: 1) Prepare the cycle graph state 퐿(푥 ) Then the lightcone 푖 of some input 2) Measure each qubit bit 푥푖 contains a distant output bit 푧푘. • in the Y basis if 푗 ∈ 푢, 푣, 푤 and 푥푗 = 1 • in the X basis otherwise The cycle relation and the HLF problem associated with a cycle 푣 The measurement pattern of the quantum algorithm is 푋/푌 identical to that used when solving one of 8 special 푋 푋 instances of the HLF 푋 푋 푋 푋 (A=adjacency matrix of cycle graph) 푢 푋/푌 푋 푋 푋 푋/푌 푤

3 In fact: Satisfying the cycle relation on input 푥 = (푥푢, 푥푣, 푥푤) ∈ {0,1} amounts to solving the associated HLF with 퐴푗,푗 = 푥푗 for 푗 ∈ {푢, 푣, 푤} We will show that quantum nonlocality beats

(A) Strictly local classical circuits 퐿(푧 ) = {푥 } 푧 (local hidden variable models) 푘 푘 for any output bit 푘

퐷 (B) Geometrically local 퐿(푥푘) ⊂ 퐵 (푥푘) for any input bit 푥푘 classical circuits in 1D

푑 (C) “Constant-depth local” |퐿(푧푘)| ≤ 퐾 for all output bits 푧푘 classical circuits Locality in constant-depth classical circuits

general shallow circuit

퐿(푧푘)

푧푘

depth 푑 = 푂 1 Locality in constant-depth classical circuits

푑 “Constant-depth locality” |퐿(푧푘)| ≤ 퐾 for all output bits 푧푘

Example: Any circuit whose depth is 푑 whose gates have fan-in ≤ 퐾.

퐿(푧푘)

푧푘

depth 푑 = 푂 1 Classical circuits solving the 2D HLF

Recall: 퐴푖,푗 = 0 unless (푖, 푗) are nearest neighbors The 2D HLF problem is the set of instances where the matrix A is the

푖 푗 퐴푖.푗 = 1 adjacency matrix of a subgraph of the 푛 × 푛 grid graph. 푖 퐴푖.푖 = 1

푛 sites

Input 퐴 Output 푛 풞푛 푧 ∈ {0,1} 푟 ∈ {0,1}ℓ Solution with probability > ퟕ/ퟖ

Suppose 푪 solves every instance of the 2D HLF problem with probability >7/8. What can be said about its locality ? Cycle relations contained in the 2D HLF

푥푖 Choose 퐴 to describe the adjacency matrix of a cycle (subgraph of the gridgraph)

푖 푗 퐴푖.푗 = 1 푢 Choose 퐴푖,푖 = 푥푖 for 푖 ∈ 푢, 푣, 푤 to 푧푘 푖 퐴푖.푖 = 푥푖 specify X or Y measurement 푣 and 퐴푗,푗= 0 for all other j

We infer (from Barrett et al.) a cycle relation satisfied by input/output: Lemma ⇒ There is an input bit 푥푖 such that 퐿(푥푖) contains a distant output bit 푧푘 Many locality constraints on 2D HLF-solving circuits 푤 푤 A classical circuit 푤 which solves the 2D HLF must satisfy all 푣 푣 ……. such cycle relations…. 푢 푣 푢 푢

…and thus satisfies constraints on the ……. lightcones of certain input bits

We show that constant-depth locality is incompatible with these constraints. 푑 |퐿(푧푘)| ≤ 퐾 for all output bits 푧푘 Quantum non-locality beats “constant-depth local” circuits Suppose a classical circuit has 푤 log(n) ≤ K fan-in and depth < 16 log(K).

Lemma: There are vertices 푢, 푣, 푤 on the even sublattice and a cycle Γ 푢 푣 passing through them such that the light cones of the input bits 푥푖 ≡ 퐴푖,푖 with 푖 ∈ {푢, 푣, 푤} do not contain any distant output bits 풛풋 ∈ 횪. Corollary (Main result): Any classical circuit which solves the 2D HLF problem with 7 log(푛) probability > must have depth . 8 >16 log(퐾) Proof of Corollary: Suppose the depth is smaller. Let Γ be as in the Lemma.

Statement for The circuit does not w.h.p solve all instances of 2D HLF geometrically local circuits problem where 퐴 is the adjacency matrix of Γ. Contradiction! Proving existence of a suitable cycle: some ingredients Suppose a classical circuit has 푤 log(n) ≤ K fan-in and depth < 16 log(K).

Lemma: There are vertices 푢, 푣, 푤 on the even sublattice and a cycle Γ 푢 푣 passing through them such that the light cones of the input bits 푥푖 ≡ 퐴푖,푖 with 푖 ∈ {푢, 푣, 푤} do not contain any distant output bits 풛풋 ∈ 횪. Proof: Based on a probabilistic argument. Proving existence of a suitable cycle: some ingredients Suppose a classical circuit has 푢 log(푛) ≤ K fan-out and depth < 16 log(퐾).

Lemma: There are vertices 푢, 푣 on the even sublattice and a path 훤 connecting them such that 푣 the light cones of the input bits 푥푖 ≡ 퐴푖,푖 with 푖 ∈ {푢, 푣} do not contain any distant output bits 풛풋 ∈ 횪. 푢 푢

푑 푑 퐿 푥푢 ≤ 퐾 and 퐿 푥푣 ≤ 퐾

푣 푣 (bounded fan-out)

퐿(푥푢) 퐿(푥푣) Proving existence of a suitable cycle: some ingredients Suppose a classical circuit has 푢 log(푛) ≤ K fan-out and depth < 16 log(퐾).

Lemma: There are vertices 푢, 푣 on the even sublattice and a path Γ connecting them such that 푣 the light cones of the input bits 푥푖 ≡ 퐴푖,푖 with 푖 ∈ {푢, 푣} do not contain any distant output bits 풛풋 ∈ 횪. 푢 푢 Example of a “good” path Γ:

no distant output bits in 푣 푣 lightcones on path

퐿(푥푢) 퐿(푥푣) Proving existence of a suitable cycle: some ingredients Suppose a classical circuit has 푢 log(푛) ≤ K fan-out and depth < 16 log(퐾).

Example of a “bad” path Γ:

distant output bits in 푣 푣 lightcones on path!

퐿(푥푢) 퐿(푥푣) Proving existence of a suitable cycle: some ingredients Suppose a classical circuit has 푢 log(푛) ≤ K fan-out and depth < 16 log(퐾).

Example of a “bad” path Γ:

distant output bits on path! 푣 푣

퐿(푥푢) 퐿(푥푣) Proving existence of a suitable cycle: some ingredients 퐵표푥(푢)

Suppose a classical circuit has 1/4 log(푛) 푛 ≤ 퐾 fan-out and depth < 16 log(퐾).

Lemma: Suppose 푢 ∈ U, v ∈ V are on the even sublattice. Then there is a path Γ connecting them such that the light cones of the input bits 푥푖 ≡ 퐴푖,푖 with 푖 ∈ {푢, 푣} 퐵표푥(푣) do not contain any distant output bits 풛풋 ∈ 횪 ∖ 푩풐풙 풖 ∪ 푩풐풙 풗 .

Pick 푢, v from certain regions: 푛/3

푛/3 U and exclude square-shaped boxes of size 푛1/4 × 푛1/4around 푢, v

V Proving existence of a suitable cycle: some ingredients 퐵표푥(푢)

Suppose a classical circuit has 1/4 log(푛) 푛 ≤ 퐾 fan-out and depth < 16 log(퐾).

1/4 Proof sketch: Boxes are of size 푛1/4 × 푛1/4 ⇒ Any pair of boxes can be connected by 푛 pairwise disjoint paths Picking a random path Γ gives 푢

퐾푑 Pr 퐿 푥 ∩ Γ ≠ ∅ ≤ → 0 for n large 푢 푛1/4 푣 풅 since 푳 풙풖 intersects at most 푲 paths.

⇒ There is a path Γ which does not intersect 퐿 푥푢 outside of 퐵표푥(푢) ∪ 퐵표푥(푣). Main result: A lower bound on classical circuits Theorem: The following holds for all sufficiently large 푛.

Input 퐴 Output 푛 풞푛 푧 ∈ {0,1} ℓ Random bits 푟 ∈ {0,1} Solution with (drawn from any probability > ퟕ/ퟖ joint distribution) For constant fan-in: Circuit must have depth 훀(퐥퐨퐠 퐧 ) On the (classical) time complexity of the HLF Hidden Linear Function (HLF) problem Input: binary symmetric matrix 퐴 Output: bit string 푧 such that 푞 푥 = 2푧푇푥 (mod 4) for all 푥 ∈ Ker(퐴)

Algorithm for general HLF: use Improved algorithm for The general HLF problem Gottesman-Knill Theorem to 2D HLF: simulate the can be solved classically simulate the quantum algorithm quantum algorithm ퟑ ퟑ in time 퐎 퐧 in time 퐎 퐧 in time 퐎 퐧ퟐ

푏1, … , 푏푘 1) Compute a basis of # used in Simulation # used in improved the nullspace Ker 퐴 circuit cost (each) circuit simulation cost (general HLF) (2D HLF) (each) 2) Solve the linear system measurement 푛 푂(푛2) 2 푧푇푏푖 = 푞(푏푖), 푖 = 1, … , 푘 푛 푂(푛) Clifford gate 푛 ≤ 푂(푛) 2 푛 푂( 푛) [S. Aaronson and D. Gottesman,PRA 70, 052328 (2004)] On the (classical) time complexity of the HLF Hidden Linear Function (HLF) problem Input: binary symmetric matrix 퐴 Output: bit string 푧 such that 푞 푥 = 2푧푇푥 (mod 4) for all 푥 ∈ Ker(퐴)

푏1, … , 푏푘 1) Compute a basis of # used in Simulation # used in improved the nullspace Ker 퐴 circuit cost (each) circuit simulation cost (general HLF) (2D HLF) (each) 2) Solve“sliding the linear window” system of measurement 푛 푂(푛2) 2 푧size푇푏푖 O(1)= 푞(푏 ×푖), 푛 푖 = 1, … , 푘 푛 푂(푛) Clifford gate 푛 ≤ 푂(푛) 2 푛 푂( 푛) [S. Aaronson and D. Gottesman,PRA 70, 052328 (2004)] Some open problems

Is this a polynomial quantum (time) speedup with constant-depth circuits?

• The quantum algorithm solves the 2D HLF Problem in time 푂(푛).

• The best-known classical algorithm takes time 푂(푛2).

Does the advantage persist if we permit stronger classical circuits?

• Can the 2D HLF be solved by 퐴퐶0 circuits? (constant depth unbounded fan-in)

Can the quantum advantage be made robust to noise?

• Different computational problems related to the HLF? arXiv:1704.00690