Quantum algorithm for P3: polar decomposition, Petz ... yquek/Slides/Petz_ ¢  Quantum algorithm

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  • Quantum algorithm for P3: polar decomposition, Petz recovery channels and pretty good measurements

    Yihui Quek

    yquek@stanford.edu

    PI QI seminar

    Joint work with various subsets of: András Gilyén, Seth Lloyd, Iman Marvian, Mark M. Wilde, Patrick Rebentrost

    October 7, 2020

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 1 / 31

  • Summary of both results

    Using Quantum Singular Value Transformation, we provide two quantum algorithms for two theoretical tools:

    1 From classical linear algebra: polar decomposition, matrix analog of z = re iθ.

    2 From quantum information: Petz recovery map, which approximately ‘reverses’ a quantum noise channel.

    Application: implementation of Pretty-Good Measurements, another ubiquitous proof tool now brought to life!

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 2 / 31

  • Summary of both results

    Using Quantum Singular Value Transformation, we provide two quantum algorithms for two theoretical tools:

    1 From classical linear algebra: polar decomposition, matrix analog of z = re iθ.

    2 From quantum information: Petz recovery map, which approximately ‘reverses’ a quantum noise channel.

    Application: implementation of Pretty-Good Measurements, another ubiquitous proof tool now brought to life!

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 2 / 31

  • Quantum Singular Value Transformation

  • Quantum Singular Value Transformation

    Block-encodings

    Unitary U is a block-encoding of A if

    U =

    [ A/α · · ·

    ] ⇐⇒ A = α(〈0|⊗s ⊗ I )U(|0〉⊗s ⊗ I ). (1)

    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) P3: Polar, Petz, PGM October 7, 2020 4 / 31

  • Quantum Singular Value Transformation

    Block-encodings

    Unitary U is a block-encoding of A if

    U =

    [ A/α · · ·

    ] ⇐⇒ A = α(〈0|⊗s ⊗ I )U(|0〉⊗s ⊗ I ). (1)

    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) P3: Polar, Petz, PGM October 7, 2020 4 / 31

  • Quantum Singular Value Transformation

    Quantum singular value transformation (I)

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

    This circuit composes U =

    [ ρ · · ·

    ] into

    [ f̃ (ρ) · · ·

    ] for your choice of polynomial f̃ .

    f̃ (ρ) means ‘apply f̃ to the singular values of ρ’.

    Usually a polynomial approximation of ideal function f .

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 5 / 31

  • Quantum Singular Value Transformation

    Quantum singular value transformation (II)

    Uρ =

    [ ρ · · ·

    ] QSVT−→

    [ f̃ (ρ) · · ·

    ] Gate complexity measured in number of uses of Uρ.

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

    e.g. Can approximate x1/2 with error 1

    2

    ∥∥∥f̃ (x)− x1/2∥∥∥ [λmin,1]

    ≤ δ

    Let κ := 1λmin(ρ) ∼ “condition number”, overall gate complexity

    O ( κ log

    1

    δ

    ) uses of Uρ.

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 6 / 31

  • Algorithm 1: Polar decomposition

  • The first P: Polar decomposition

    Polar decomposition

    Definition (Polar decomposition)

    The polar decomposition of A ∈ CM×N is the factorization

    A = UB = B̃U

    where U is a unitary/isometry and B = √ A†A, B̃ =

    √ AA† Hermitian.

    Task: enact U := Upolar(A) on an input quantum state.

    |ψ〉 → NUpolar(A)|ψ〉

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 8 / 31

  • The first P: Polar decomposition

    Geometrical intuition for the polar decomposition

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 9 / 31

  • The first P: Polar decomposition

    Geometrical intuition for the polar decomposition

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 9 / 31

  • The first P: Polar decomposition

    Application of polar decomposition: Procrustes problem

    Given: r (input, output) quantum state pairs: (|φi 〉, |ψi 〉)ri=1. Which U best transforms each input to output?

    Figure 1: Taken from http://atlasgeographica.com/the-bed-of-procrustes/

    Solution: U∗ is the polar decomposition of FG †, where F has |φi 〉 as columns and G has |ψi 〉 as columns.

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 10 / 31

    http://atlasgeographica.com/the-bed-of-procrustes/

  • The first P: Polar decomposition

    Our algorithm

    [Lloyd’20] provided an algorithm for enacting Upolar(A) based on density matrix exponentiation and quantum phase estimation.

    QSVT simplifies this dramatically: One realizes that

    SVD(A) = r∑

    i=1

    σii |pi 〉〈qi | ⇒ Upolar(A) = r∑

    i=1

    |pi 〉〈qi |.

    i.e. can simply use QSVT to set all non-zero singular values to 1.

    Significant speedups

    vs [Lloyd ’20]: exponentially faster in ε, with polynomial speedups in r , κ.

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 11 / 31

  • Algorithm 2: Petz recovery map

  • The second P: Petz recovery map

    arxiv:2006.16924

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 13 / 31

  • The second P: Petz recovery map Background and 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) P3: Polar, Petz, PGM October 7, 2020 14 / 31

  • The second P: Petz recovery map Background and 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) . (2)

    Quantumly: Petz recovery map! Given a forward channel, N and an ‘implicit’ input state σA:

    Pσ,NB→A(·) := σ 1/2 A N †

    ( N (σA)−1/2(·)N (σA)−1/2

    ) σ

    1/2 A (3)

    Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 15 / 31

  • The second P: Petz recovery map Background and intuition for the Petz map

    Why should you care about the Petz map?

    1 Universal recovery operation in error correction [Barnum-Knill’02, Ng-Mandayam’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) P3: Polar, Petz, PGM October 7, 2020 16 / 31

  • The second P: Petz recovery map Background and intuition for the Petz map

    Why should you care about the Petz map?

    1 Universal recovery operation in error correction [Barnum-Knill’02, Ng-Mandayam’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) P3: Polar, Petz, PGM October 7, 2020 16 / 31

  • The second P: Petz recovery map Background and intuition for the Petz map

    Why should you care about the Petz map?

    1 Universal recovery operation in error correction [Barnum-Knill’02, Ng-Mandayam’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) P3: Polar, Petz, PGM October 7, 2020 16 / 31

  • The second P

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