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 Background 1 Intuition for the Petz map The Petz map in Physics QI crash course The quantum singular value transform 2 Block-encodings QSVT Our algorithm 3 Assumptions Re-writing the map Steps Application: Pretty-Good Measurements 4 Optimality 5 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 p X ( x ) and channel p Y | X ( y | x ), what is p X | 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’: p X | Y ( x | y ) = p X ( x ) p Y | X ( y | x ) . (1) p Y ( y ) Quantumly : Petz recovery map! Given a forward channel, N and an input state σ A : A N † � N ( σ A ) − 1 / 2 ( · ) N ( σ A ) − 1 / 2 � P σ, N B → A ( · ) := σ 1 / 2 σ 1 / 2 (2) A 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 A R ∗ AB φ (3) B A wild Petz map has appeared in quan- φ (1) ab ab R ∗ AB tum gravity ! R ∗ φ (2) 3 [Cotler-Hayden-Penington- BC bc φ (4) bc Salton-Swingle-Walter ’18] D R ∗ BC C 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 A R ∗ AB φ (3) B A wild Petz map has appeared in quan- φ (1) ab ab R ∗ AB tum gravity ! R ∗ φ (2) 3 [Cotler-Hayden-Penington- BC bc φ (4) bc Salton-Swingle-Walter ’18] D R ∗ BC C 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 A R ∗ AB φ (3) B A wild Petz map has appeared in quan- φ (1) ab ab R ∗ AB tum gravity ! R ∗ φ (2) 3 [Cotler-Hayden-Penington- BC bc φ (4) bc Salton-Swingle-Walter ’18] D R ∗ BC C 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 N A → B ( · ) can be decomposed as d − 1 � V l X A V † N A → B ( X A ) = l l =0 where V l are linear (‘Kraus’) operators and � d − 1 l =0 V † l V l = I A . (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 : UU † = U † U = I . U ( ρ ) = U ρ U † Definition (Channel adjoint) Given N A → B , the channel adjoint N † B → A satisfies � Y , N ( X ) � = �N † ( Y ) , X � ∀ X ∈ H A , Y ∈ H B (Explicitly, with Kraus op.s: N † ( Y ) = � d − 1 l =0 V † l YV l .) 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 N A → B , a unitary extension U : H A ⊗ H E → H B ⊗ H E ′ of N satisfies Tr E ′ ( U ( ρ ⊗ | 0 �� 0 | E ) U † ) = N A → 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 Background 1 Intuition for the Petz map The Petz map in Physics QI crash course The quantum singular value transform 2 Block-encodings QSVT Our algorithm 3 Assumptions Re-writing the map Steps Application: Pretty-Good Measurements 4 Optimality 5 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 � A /α � · A = α ( � 0 | ⊗ s ⊗ I ) U ( | 0 � ⊗ s ⊗ I ) . U = ⇐ ⇒ (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 � A /α � · A = α ( � 0 | ⊗ s ⊗ I ) U ( | 0 � ⊗ s ⊗ I ) . U = ⇐ ⇒ (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 σ A (implicit ‘input’ to channel ∼ p X ), quantum channel N A → B , Petz map is: A N † � N ( σ A ) − 1 / 2 ω B N ( σ A ) − 1 / 2 � P σ, N B → A ( ω B ) := σ 1 / 2 σ 1 / 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 σ A (implicit ‘input’ to channel ∼ p X ), quantum channel N A → B , Petz map is: A N † � N ( σ A ) − 1 / 2 ω B N ( σ A ) − 1 / 2 � P σ, N B → A ( ω B ) := σ 1 / 2 σ 1 / 2 A , Composition of 3 CP maps (overall trace-preserving): ( · ) → [ N ( σ A )] − 1 / 2 ( · ) [ N ( σ A )] − 1 / 2 ( · ) → N † ( · ) , ( · ) → σ 1 / 2 A ( · ) σ 1 / 2 A . Will need to block-encode σ A . 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 U ψ that prepares a purification | ψ σ � : � σ A � · RA ) † ( I R ⊗ SWAP AA ′ ) U ψ ( U ψ RA = (5) · · Exact block-encoding with two uses of U ψ . Yihui Quek (Stanford) Q. Algo for Petz map and PGood meas. July 31, 2020 14 / 31
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