New Hardness Results for the Permanent Using Linear Optics Daniel - - PowerPoint PPT Presentation

New Hardness Results for the Permanent Using Linear Optics

Daniel Grier Luke Schaeffer MIT

Permanent Review

Permanent: Given matrix Example:

Permanent complexity

Theorem [Valiant (1979)]: The permanent of a matrix is -hard to compute.

#P

• hardness: Let be an oracle which computes the permanent of a matrix.

#P ⊆ FPper

per

Question: Can the permanent be efficiently computed? Probably not: collapses

per ∈ PH = ⇒ PH

Ryser’s/Glynn’s formula: Permanent can be computed in time .

Permanent counts cycle covers

Combinatorial interpretation:

• If is adjacency matrix, then

= the number of cycle covers of graph.

A

per(A)

• If is adjacency matrix with edge weights, then

= the sum of the weighted cycle covers of graph.

A

per(A)

Valiant’s reduction

Idea behind Valiant’s proof: Construct graph such that the weighted cycle covers correspond to the number of solutions to a 3SAT formula. Figure: A single “interchange” in Valiant’s original proof

Going beyond Valiant’s reduction

Drawbacks to Valiant’s reduction: 1) Relies on complicated cycle cover gadgets

• Terry Rudolph (2009): Built subclass of quantum circuits with amplitudes

proportional to the permanent

• Ben-Dor and Halevi (1993): Simplified cycle cover argument
• Scott Aaronson (2011): #P-hardness of permanent from linear optics

Why Quantum? Offload difficulty onto well-known theorems in linear optics 2) Not suited for “structured” matrices

• Invertible: Valiant’s matrices are probably invertible, but tedious to prove
• Unitary: Valiant’s matrices are not unitary, and no obvious way forward

Plan: Modify Aaronson’s proof and use quantum reductions to handle classes of matrices not suited for reductions based on cycle covers.

#P-hardness for new classes of matrices

Theorem: The permanent of an matrix in any of the classical Lie groups over the complex numbers is -hard:

n × n

A

#P

Theorem: Let be prime. There exists a finite field of characteristic , namely , such that the permanent of an orthogonal matrix in is hard for the class .

p 6= 2, 3

p

Fp4

Fp4

ModpP

Dichotomy

p = 2

: Permanent = determinant

p = 3

: Nontrivial algorithm due to Kogan (1996) Theorem: The permanent of an orthogonal matrix over is -hard for 1/16th of all primes.

Fp

ModpP

Outline of Aaronson’s proof

Input: Given polynomially sized circuit Output: Number of unsatisfying assignments minus satisfying assignments to

C : {0, 1}n → {0, 1}

C

∆C := X

x∈{0,1}n

(−1)C(x)

2 Convert into a linear

• ptical network

Q

L

3 Use correspondence between linear optics and permanents

/ h1, 0, . . .| ϕ(L) |1, 0, . . .i

∝ per(LI,I)

• ur contribution

Encode into the transition amplitude of a quantum circuit

• ver qubits

1

∆C

Q

∆C / h0 . . . 0| Q |0 . . . 0i

Comparison of linear optics

Quantum computing with qudits Linear optics with photons States:

|ψi 2 (Cm)⊗n

|ψi 2 (Cm)n

States: symmetric tensor product

v1 . . . vn = 1 n! X

σ∈Sn

vσ(1) ⌦ . . . ⌦ vσ(n)

Operations: U ∈ U(mn)

Operations: for

L⊗n

L ∈ U(m)

Linear Optics - States

States: photons and modes

n

m

photons ~ indistinguishable balls modes ~ distinct bins/locations

s2

Notation: Let be the state with photons in the first mode, in the second, and so on.

|s1, s2, . . . , smi

s1

For example: |1, 1, 1, 0, 0, 0i + |0, 2, 0, 0, 1, 0i

p 2

Linear Optics - Transformations

where is the matrix obtained by taking

• copies of row from
• copies of column from
• transition formula: Given unitary , the amplitude from state

to state is |Si = |s1, s2, . . . , smi

|Ti = |t1, t2, . . . , tmi

ϕ

m × m

L

hT| ϕ(L) |Si = per(LS,T ) ps1!s2! . . . sm!t1!t2! . . . tm!

LS,T

si

ti

i

i

L

L

Observation: If , then .

|Si = |Ti = |1, . . . , 1i

hT| ϕ(L) |Si = per(L)

Idea: Linear optical transformation is specified by its action on a single photon. Apply homomorphism to lift to entire Hilbert space for multiple photons.

L = 1 √ 2 ✓ 1 1 1 −1 ◆

Example:

LS,T = 1 √ 2 ✓1 1 1 1 ◆

h2, 0| ϕ(L) |1, 1i = 1 p 2

|Si = |1, 1i

|Ti = |2, 0i

Outline of #P-hardness proof

Input: Given polynomially sized circuit Output: Number of satisfying assignments minus unsatisfying assignments to

C : {0, 1}n → {0, 1}

C

∆C := X

x∈{0,1}n

(−1)C(x)

Encode into the transition amplitude of a quantum circuit

• ver qubits

1

∆C

2 Convert into a linear

• ptical network

Q

L

3 Use correspondence between linear optics and permanents

Q

1 3

Postselected linear optics is quantum universal

Theorem [Aaronson (2011)]:

∆C 2n = h0 . . . 0| Q |0 . . . 0i = 4Γ hI| ϕ(L) |Ii = 4Γper(LI,I)

not unitary

hI| ϕ(L) |Ii = 1 4Γ h0 · · · 0| Q |0 · · · 0i

where,

Γ =

number of CSIGN gates in Q

Theorem [Knill, Laflamme, Milburn (2001)]: Postselected linear optical circuits are universal for quantum computation. Formally, given quantum circuit with polynomially many CSIGN and single- qubit gates, there exists linear optical circuit with polynomially many modes such that

Q

L

|Ii = |0, 1, 0, 1, . . . , 0, 1i

Note: CSIGN + single-qubits gates are universal for quantum computation

CSIGN |x1x2i = (1)x1x2 |x1x2i

KLM protocol - representing states

hI| ϕ(L) |Ii = 1 4Γ h0 · · · 0| Q |0 · · · 0i

where Theorem [Knill, Laflamme, Milburn (2001)]: Given quantum circuit with polynomially many CSIGN and single-qubit gates, there exists linear optical circuit with polynomially many modes such that

Q

L

Representing qubits with linear optical states:

|Ii = |0, 1, 0, 1, . . . , 0, 1i

This is the source of non-unitarity in Aaronson’s proof Solution: use two modes and one photon to encode a single qubit

|0i ! |0, 1i

|1i ! |1, 0i

qubits linear optical state Dual rail encoding Problem: qubit is either in state or , but number of photons is conserved

|0i

|1i

Add new encoding phase to KLM

Problem: KLM uses dual rail encoding. Solution: Prepare the dual rail encoding using another gadget. KLM solution: 1 qubit represented by 1 photon and 2 modes Our solution: 1 qubit represented by 4 photons and 4 modes

|1, 1, 1, 1i

|0, 1, 2, 1i

Encoding gadget dual rail encoding dumpster postselected photon Goal: Construct linear optical circuit from such that

L

Q

h1, 1, . . . , 1| ϕ(L) |1, 1, . . . , 1i / h0 · · · 0| Q |0 · · · 0i

Add new encoding phase to KLM

Problem: KLM uses dual rail encoding. Solution: Prepare the dual rail encoding using another gadget. KLM solution: 1 qubit represented by 1 photon and 2 modes Our solution: 1 qubit represented by 4 photons and 4 modes

|1, 1, 1, 1i

|0, 1, 2, 1i

Encoding gadget

|1, 1, 1, 1i

Decoding gadget Goal: Construct linear optical circuit from such that

L

Q

h1, 1, . . . , 1| ϕ(L) |1, 1, . . . , 1i / h0 · · · 0| Q |0 · · · 0i

Putting it all together

Theorem: unitary

∆C 2n = h0 . . . 0| Q |0 . . . 0i

= ( p 6)n 3 r 3 2 !Γ h1, . . . , 1| ϕ(L) |1, . . . , 1i

= (− √ 6)n 3 r 3 2 !Γ per(L)

E = 1 √ 6   √ 2 − √ 2 √ 2 √ 3 √ 3 −2 −1 1  

How do you find gadgets?

• 1. Guess transformation
• 2. Use constraint solver

Permanent hardness over finite fields

Theorem: Permanent is -hard for unitary matrices. Theorem: Let be prime. There exists a finite field of characteristic , namely , such that the permanent of an orthogonal matrix in is - hard.

p 6= 2, 3

p

Fp4

Fp4

ModpP

E = 1 √ 6   √ 2 − √ 2 √ 2 √ 3 √ 3 −2 −1 1  

Proof: Inspect gadgets carefully

V = 1 3 √ 2 B B B @ − √ 2 −2 2 2 √ 2 2 − √ 2 −2 √ 2 2 − p 6 + 2 √ 6 p 6 − 2 √ 6 − p 3 + √ 6 p 3 − √ 6 − p 6 − 2 √ 6 − p 6 + 2 √ 6 − p 3 − √ 6 − p 3 + √ 6 1 C C C A

#P

All entries in Q(α)

α = q 2 + √ 2 + q 3 + √ 6

Summarizing matrix permanent complexity

Cn×n

SO(n)

{0, 1}n×n

xT Ax ≥ 0

exact #P-hard [Valiant 79] #P-hard [GS 2017] #P-hard [Valiant 79] #P-hard [GS 2017] approximate #P-hard [Valiant 79] #P-hard [GS 2017] FPTAS [JSV 2004] ???

Open Problems:

• Is there a polynomial-time approximation algorithm for permanents of

positive-semidefinite matrices?

• best known: polynomial time -approximation [AGGS 2017]

4.84n

• Are orthogonal permanents over hard for for all ?

Fp

ModpP

p 6= 2, 3

• Are there more insights about the permanent to be gained through

this linear optical lens?

• [CCG 2016] : under restricted conditions on the eigenvalues, can
• utperform Gurvits’s additive approximation algorithm