Quantum algorithm for Petz recovery channels and pretty good - - PowerPoint PPT Presentation

Quantum algorithm for Petz recovery channels and pretty good measurements (arXiv:2006.16924)

Yihui Quek

yquek@stanford.edu MIT QIS group meeting with Andr´ as Gily´ en, Seth Lloyd, Iman Marvian, Mark M. Wilde

July 31, 2020

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 1 / 31

arxiv:2006.16924

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 2 / 31

Our results

The Petz recovery map approximately ’reverses’ a known quantum noise channel and is ubiquitous as a theoretical tool. Yet no systematic implementation exists! Using the Quantum Singular Value Transform toolbox, we provide such a systematic implementation. Consequence: can also perform Pretty-Good Measurements, a common proof tool in algorithms.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 3 / 31

Background

Outline

1

Background Intuition for the Petz map The Petz map in Physics QI crash course

2

The quantum singular value transform Block-encodings QSVT

3

Our algorithm Assumptions Re-writing the map Steps

4

Application: Pretty-Good Measurements

5

Optimality

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 4 / 31

Background Intuition for the Petz map

Classical ‘reversal’ channel from Bayes’ theorem

Given input pX(x) and channel pY |X(y|x), what is pX|Y (x|y)?

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 5 / 31

Background Intuition for the Petz map

Petz recovery

Classically, Bayes theorem yields ‘reverse channel’: pX|Y (x|y) = pX(x)pY |X(y|x) pY (y) . (1) Quantumly: Petz recovery map! Given a forward channel, N and an input state σA: Pσ,N

B→A(·) := σ1/2 A N †

N(σA)−1/2(·)N(σA)−1/2 σ1/2

A

(2)

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 6 / 31

Background The Petz map in Physics

Why should you care about the Petz map?

1 Universal recovery operation in error correction [Barnum-Knill’02,

Ng-Mandayam’09, Tyson’09]

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 7 / 31

Background The Petz map in Physics

Why should you care about the Petz map?

1 Universal recovery operation in error correction [Barnum-Knill’02,

Ng-Mandayam’09, Tyson’09]

2 Important proof tool in QI: [Beigi-Datta-Leditzky’16] as a decoder

in quantum communication, achieves coherent information rate

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 7 / 31

Background The Petz map in Physics

Why should you care about the Petz map?

1 Universal recovery operation in error correction [Barnum-Knill’02,

Ng-Mandayam’09, Tyson’09]

2 Important proof tool in QI: [Beigi-Datta-Leditzky’16] as a decoder

in quantum communication, achieves coherent information rate

3

A wild Petz map has appeared in quan- tum gravity!

[Cotler-Hayden-Penington- Salton-Swingle-Walter ’18]

φ(1)

ab

φ(2)

bc

φ(3)

ab

φ(4)

bc

R∗

AB

R∗

AB

R∗

BC

R∗

BC

A B C D

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 7 / 31

Background The Petz map in Physics

Why should you care about the Petz map?

1 Universal recovery operation in error correction [Barnum-Knill’02,

Ng-Mandayam’09, Tyson’09]

2 Important proof tool in QI: [Beigi-Datta-Leditzky’16] as a decoder

in quantum communication, achieves coherent information rate

3

A wild Petz map has appeared in quan- tum gravity!

[Cotler-Hayden-Penington- Salton-Swingle-Walter ’18]

φ(1)

ab

φ(2)

bc

φ(3)

ab

φ(4)

bc

R∗

AB

R∗

AB

R∗

BC

R∗

BC

A B C D

4 Is a type of quantum “Bayesian inference” [Leifer-Spekkens’13]

(see: ⋆-product)?

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 7 / 31

Background The Petz map in Physics

Why should you care about the Petz map?

1 Universal recovery operation in error correction [Barnum-Knill’02,

Ng-Mandayam’09, Tyson’09]

2 Important proof tool in QI: [Beigi-Datta-Leditzky’16] as a decoder

in quantum communication, achieves coherent information rate

3

A wild Petz map has appeared in quan- tum gravity!

[Cotler-Hayden-Penington- Salton-Swingle-Walter ’18]

φ(1)

ab

φ(2)

bc

φ(3)

ab

φ(4)

bc

R∗

AB

R∗

AB

R∗

BC

R∗

BC

A B C D

4 Is a type of quantum “Bayesian inference” [Leifer-Spekkens’13]

(see: ⋆-product)?

5 Has pretty-good measurements as a special case (later) Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 7 / 31

Background QI crash course

Quantum channels I

Quantum channel (informal): a physically valid map bringing one quantum state to another. Important use case: model for quantum noise, e.g. amplitude damping channels Theorem (Choi-Kraus theorem) Any physically valid channel NA→B(·) can be decomposed as NA→B(XA) =

d−1

• l=0

VlXAV †

l

where Vl are linear (‘Kraus’) operators and d−1

l=0 V † l Vl = IA.

(and vice versa!)

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 8 / 31

Background QI crash course

Quantum channels II

The Kraus operator representation is very useful. Example (Unitary evolution) Has a single Kraus operator U: U(ρ) = UρU† UU† = U†U = I. Definition (Channel adjoint) Given NA→B, the channel adjoint N †

B→A satisfies

Y , N(X) = N †(Y ), X ∀X ∈ HA, Y ∈ HB (Explicitly, with Kraus op.s: N †(Y ) = d−1

l=0 V † l YVl.)

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 9 / 31

Background QI crash course

Unitary extensions

Every channel can be replicated by a unitary acting on a larger input. Definition (Unitary extension) Given a channel NA→B , a unitary extension U : HA ⊗ HE → HB ⊗ HE ′

• f N satisfies

TrE ′(U(ρ ⊗ |00|E)U†) = NA→B(ρ) (3) Q: How big does the environment need to be? A: Dimension at least the number of Kraus operators, d.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 10 / 31

The quantum singular value transform

Outline

1

Background Intuition for the Petz map The Petz map in Physics QI crash course

2

The quantum singular value transform Block-encodings QSVT

3

Our algorithm Assumptions Re-writing the map Steps

4

Application: Pretty-Good Measurements

5

Optimality

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 11 / 31

The quantum singular value transform Block-encodings

Block-encodings

Unitary U is a block-encoding of A if U = A/α · · ·

• ⇐

⇒ A = α(0|⊗s ⊗ I)U(|0⊗s ⊗ I). (4) U (acts on a qubits +s ancillae) can be used to realize a probabilistic implementation of A/α.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 12 / 31

The quantum singular value transform Block-encodings

Block-encodings

Unitary U is a block-encoding of A if U = A/α · · ·

• ⇐

⇒ A = α(0|⊗s ⊗ I)U(|0⊗s ⊗ I). (4) U (acts on a qubits +s ancillae) can be used to realize a probabilistic implementation of A/α. On a-qubit input |ψ, Apply U to |0⊗s ⊗ |ψ Measure ancillae; if outcome was |0⊗s, the first a qubits contain a state ∼ A|ψ.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 12 / 31

The quantum singular value transform Block-encodings

A peek at the Petz recovery map

Given: quantum state σA (implicit ‘input’ to channel ∼ pX), quantum channel NA→B, Petz map is: Pσ,N

B→A(ωB) := σ1/2 A N †

N(σA)−1/2ωBN(σA)−1/2 σ1/2

A ,

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 13 / 31

The quantum singular value transform Block-encodings

A peek at the Petz recovery map

Given: quantum state σA (implicit ‘input’ to channel ∼ pX), quantum channel NA→B, Petz map is: Pσ,N

B→A(ωB) := σ1/2 A N †

N(σA)−1/2ωBN(σA)−1/2 σ1/2

A ,

Composition of 3 CP maps (overall trace-preserving): (·) → [N(σA)]−1/2 (·) [N(σA)]−1/2 (·) → N †(·), (·) → σ1/2

A (·)σ1/2 A .

Will need to block-encode σA.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 13 / 31

The quantum singular value transform Block-encodings

How to block-encode a density matrix?

Depends on how the density matrix σ is provided: Physical copies of σ: use density matrix exponentiation [Lloyd-Mohseni-Rebentrost’13] → approximate block-encoding Access to circuit Uψ that prepares a purification |ψσ: (Uψ

RA)† (IR ⊗ SWAPAA′) Uψ RA =

σA · · ·

• (5)

Exact block-encoding with two uses of Uψ.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 14 / 31

The quantum singular value transform QSVT

Quantum singular value transformation (I)

QSVT: A method to transform singular values of block-encodings [Gily´ en-Su-Low-Wiebe’18, Low-Chuang ’16]

This circuit implements U = ρ · · ·

• QSVT

− → ˜ f (ρ) · · ·

• ˜

f is a function applied to the singular values of its argument. Usually a polynomial approximation of ideal function f .

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 15 / 31

The quantum singular value transform QSVT

Quantum singular value transformation (II)

Uρ = ρ · · ·

• QSVT

− → ˜ f (ρ) · · ·

• Gate complexity measured in number of uses of Uρ.

Depends on approximation’s domain ([θ, 1]) and error (δ). Approximation error: 1 2

• ˜

f (x) − x1/2

• [λmin,1] ≤ δ

Define κ :=

1 λmin(ρ) ∼ “condition number”, overall

O

• κ log 1

δ

• uses of Uρ.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 16 / 31

Our algorithm

Outline

1

Background Intuition for the Petz map The Petz map in Physics QI crash course

2

The quantum singular value transform Block-encodings QSVT

3

Our algorithm Assumptions Re-writing the map Steps

4

Application: Pretty-Good Measurements

5

Optimality

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 17 / 31

Our algorithm Assumptions

Assumptions

We have the following quantum circuits:

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 18 / 31

Our algorithm Assumptions

Assumptions

We have the following quantum circuits:

1 Block-encodings of two states

The implicit state σA (Uσ = σA · · ·

• )

The state N(σA) (UN (σ) = N(σA) · · ·

• ).

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 18 / 31

Our algorithm Assumptions

Assumptions

We have the following quantum circuits:

1 Block-encodings of two states

The implicit state σA (Uσ = σA · · ·

• )

The state N(σA) (UN (σ) = N(σA) · · ·

• ).

2 UN

E ′A→EB, a unitary extension of the forward channel N

Setting: we have characterized the noise and can simulate it using quantum gates.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 18 / 31

Our algorithm Re-writing the map

Re-writing the channel adjoint

Second step of map: (·) → N †(·) Can write adjoint N † in terms of unitary extension UN (which we assumed we could implement): N †(ωB) = 0|E ′UN † (IE ⊗ ωB) UN |0E ′

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 19 / 31

Our algorithm Re-writing the map

Re-writing the channel adjoint

Second step of map: (·) → N †(·) Can write adjoint N † in terms of unitary extension UN (which we assumed we could implement): N †(ωB) = 0|E ′UN † (IE ⊗ ωB) UN |0E ′ Problem: IE is not a quantum state. Solution: act on maximally-entangled state

1 dE

dE −1

i=0

|iE|i ˜

E, whose

density matrix is ∼ identity.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 19 / 31

Our algorithm Re-writing the map

Overview of algorithm

Will implement an isometric extension of the Petz map:

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 20 / 31

Our algorithm Re-writing the map

Overview of algorithm

Will implement an isometric extension of the Petz map: I) (I) [N(σA)]−1/2 (·) [N(σA)]−1/2

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 20 / 31

Our algorithm Re-writing the map

Overview of algorithm

Will implement an isometric extension of the Petz map: II) I) (I) [N(σA)]−1/2 (·) [N(σA)]−1/2 (II) N †(·)

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 20 / 31

Our algorithm Re-writing the map

Overview of algorithm

Will implement an isometric extension of the Petz map: II) I) III) (I) [N(σA)]−1/2 (·) [N(σA)]−1/2 (II) N †(·) (III) σ1/2

A (·)σ1/2 A .

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 20 / 31

Our algorithm Re-writing the map

Overview of algorithm

Will implement an isometric extension of the Petz map: II) I) III) (I) [N(σA)]−1/2 (·) [N(σA)]−1/2 (II) N †(·) (III) σ1/2

A (·)σ1/2 A .

Finally: Tracing over environment ˜ E then implements the map.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 20 / 31

Our algorithm Re-writing the map

Theorem For a forward channel N and an implicit input state σA, we can realize an approximation ˜ P of the associated Petz recovery channel P, such that:

• ˜

PσA,N − PσA,N

• ⋄ ≤ ε,

(6) with

• O
• dEκN(σ)
• uses of UN (channel unitary)

(7)

• O
• poly(dE, κN(σ), κ(σ))
• uses of UσA and UN(σA)

(8) dE is the dimension of the system E, which is at least the Kraus rank of the channel N(·).

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 21 / 31

Our algorithm Steps

Maps I and III

Maps I and III are a matter of transforming block-encodings: I) III) Map I: UN(σ) = N(σA) · · ·

• QSVT

− →

• N(σA)−1/2

· · ·

• using

˜ f1(x) ≈ x−1/2 with ˜ O

• κN(σ)
• uses of UN(σ).

Map III: Uσ = σA · · ·

• QSVT

− →

• σ1/2

A

· · ·

• using ˜

f2(x) ≈ x1/2 with ˜ O(κσ) uses of Uσ.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 22 / 31

Our algorithm Steps

Map II: Channel adjoint N(·)

1 Tensor in the maximally entangled state ΓE ˜

E/dE

2 Perform UN †.

Easy peasy, no QSVT needed.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 23 / 31

Our algorithm Steps

Map II: Channel adjoint N(·)

3 Measure the system E ′, accepting if the all-zeros outcome occurs. 4 Ignore the system ˜

E (i.e. trace it out).

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 23 / 31

Our algorithm Steps

Map II: Channel adjoint N(·)

3 Measure the system E ′, accepting if the all-zeros outcome occurs. 4 Ignore the system ˜

E (i.e. trace it out). Not contiguous with steps 1 and 2 as there’s an intervening σ

1 2

A →

implement 3 and 4 after Map III.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 23 / 31

Our algorithm Steps

More on measurement

Implementing in sequence the unitaries created through SVT obtains the

• verall unitary

˜ W =

• 1

4

• 1

dE κN (σ) ˜

V · · ·

• (9)

where V = ideal Petz map, ˜ V − V < O(ε). This is a probabilistic implementation: Measuring E ′ system, probability psuccess = O( 1

dE κ) of getting |0.

Make this deterministic: use Oblivious Amplitude Amplification to boost probability by repeating ˜ W O

• 1/√psuccess
• times .

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 24 / 31

Application: Pretty-Good Measurements

Outline

1

Background Intuition for the Petz map The Petz map in Physics QI crash course

2

The quantum singular value transform Block-encodings QSVT

3

Our algorithm Assumptions Re-writing the map Steps

4

Application: Pretty-Good Measurements

5

Optimality

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 25 / 31

Application: Pretty-Good Measurements

Pretty-Good Measurements

Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, we are promised that ρ is in state σx with probability p(x). What POVM maximizes Pr(correctly identify ρ)?

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 26 / 31

Application: Pretty-Good Measurements

Pretty-Good Measurements

Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, we are promised that ρ is in state σx with probability p(x). What POVM maximizes Pr(correctly identify ρ)? No optimal strategy when |X| ≥ 3, but ‘pretty-good measurement’ does pretty-well on this. [Hausladen-Wootters’94]

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 26 / 31

Application: Pretty-Good Measurements

Pretty-Good Measurements

Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, we are promised that ρ is in state σx with probability p(x). What POVM maximizes Pr(correctly identify ρ)? No optimal strategy when |X| ≥ 3, but ‘pretty-good measurement’ does pretty-well on this. [Hausladen-Wootters’94] Special case of the Petz map with σXB =

x pX(x)|x

x|X ⊗ σx

B, and

N the partial trace over X.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 26 / 31

Application: Pretty-Good Measurements

Pretty-Good Measurements

Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, we are promised that ρ is in state σx with probability p(x). What POVM maximizes Pr(correctly identify ρ)? No optimal strategy when |X| ≥ 3, but ‘pretty-good measurement’ does pretty-well on this. [Hausladen-Wootters’94] Special case of the Petz map with σXB =

x pX(x)|x

x|X ⊗ σx

B, and

N the partial trace over X. Our algorithm can implement this, with O

• |X|poly(κ)
• uses of

unitary preparing σXB.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 26 / 31

Application: Pretty-Good Measurements

Pretty-Good Measurements

Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, we are promised that ρ is in state σx with probability p(x). What POVM maximizes Pr(correctly identify ρ)? No optimal strategy when |X| ≥ 3, but ‘pretty-good measurement’ does pretty-well on this. [Hausladen-Wootters’94] Special case of the Petz map with σXB =

x pX(x)|x

x|X ⊗ σx

B, and

N the partial trace over X. Our algorithm can implement this, with O

• |X|poly(κ)
• uses of

unitary preparing σXB.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 26 / 31

Application: Pretty-Good Measurements

Why should you care about Pretty-Good Measurements?

1 Used to approach the Holevo information rate

[Hausladen-Jozsa-Schumacher-Westmoreland-Wootters’96]

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 27 / 31

Application: Pretty-Good Measurements

Why should you care about Pretty-Good Measurements?

1 Used to approach the Holevo information rate

[Hausladen-Jozsa-Schumacher-Westmoreland-Wootters’96]

2 Important proof technique for q. algorithms: PGM ∼ optimal

measurement to distinguish states

i

Is the optimal measurement in q. algorithm for dihedral hidden subgroup problem [Bacon-Childs-vanDam’06]

ii

Bounds on sample complexity for Quantum Probably Approximately Correct (PAC) learning [Arunachalam-de Wolf’16]

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 27 / 31

Optimality

Outline

1

Background Intuition for the Petz map The Petz map in Physics QI crash course

2

The quantum singular value transform Block-encodings QSVT

3

Our algorithm Assumptions Re-writing the map Steps

4

Application: Pretty-Good Measurements

5

Optimality

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 28 / 31

Optimality

Is O dEκN(σ)

• uses of UN optimal?

Claim Any algorithm whose gate complexity depends on dE or κN(σ) must use UN at least Ω

• dα

E κβ N(σ)

• times, for α + β ≥ 1

2.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 29 / 31

Optimality

Is O dEκN(σ)

• uses of UN optimal?

Claim Any algorithm whose gate complexity depends on dE or κN(σ) must use UN at least Ω

• dα

E κβ N(σ)

• times, for α + β ≥ 1

2.

Proof. (Idea) Will construct explicit example (∼ Grover’s search) where κN(σ) = dE = N and √ N uses of UN necessary.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 29 / 31

Optimality

Search-based optimality argument

Proof. Setting: N elements, one marked. Oracle O recognizes marked element by flagging with |1. Forward channel N: applies O and outputs its output. Implicit input σA: uniformly random index in [N]. Petz(N, σA): applied on |11|, finds the marked element with

• certainty. i.e. Applying Petz does unstructured search!

But according to Grover’s search lower bound, this requires Ω( √ N) = Ω

• d

1 2 −α

E

κα

N(σ)

• uses of O.

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 30 / 31

Optimality

Takeaways!

Our algorithm for Petz recovery maps using the quantum singular value transform Is systematic and rigorous Also allows to implement Pretty-Good Measurements (q: what other measurements?) Is pretty close to optimal in gate complexity Brings these theoretical tools closer to implementation on an error-corrected quantum computer

Find us at arxiv:2006.16924!

Yihui Quek (Stanford)

• Q. Algo for Petz map and PGood meas.

July 31, 2020 31 / 31