Quantum algorithm for P3: polar decomposition, Petz...
Transcript of Quantum algorithm for P3: polar decomposition, Petz...
Quantum algorithm for P3: polar decomposition, Petzrecovery channels and pretty good measurements
Yihui Quek
PI QI seminar
Joint work with various subsets of: Andras Gilyen, 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 twoquantum algorithms for two theoretical tools:
1 From classical linear algebra: polar decomposition, matrix analog ofz = 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 twoquantum algorithms for two theoretical tools:
1 From classical linear algebra: polar decomposition, matrix analog ofz = 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 ValueTransformation
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 probabilisticimplementation 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 astate ∼ 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 probabilisticimplementation 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 astate ∼ 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[Gilyen-Su-Low-Wiebe’18, Low-Chuang ’16]
This circuit composes U =
[ρ ·· ·
]into
[f (ρ) ·· ·
]for your choice of polynomial f .
f (ρ) means ‘apply f 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−→
[f (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 error1
2
∥∥∥f (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 = BU
where U is a unitary/isometry and B =√A†A, B =
√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 Ubest 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
The first P: Polar decomposition
Our algorithm
[Lloyd’20] provided an algorithm for enacting Upolar(A) based ondensity 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(·) := σ1/2A N †
(N (σA)−1/2(·)N (σA)−1/2
)σ
1/2A (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 decoderin 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 decoderin 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 decoderin 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 decoderin 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 decoderin 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 QIT crash course
Quantum channels I
Quantum channel (informal): a physically valid map bringing onequantum state to another.
Important use case: model for quantum noise, e.g. amplitudedamping 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) P3: Polar, Petz, PGM October 7, 2020 17 / 31
The second P: Petz recovery map QIT crash course
Quantum channels II
Definition (Channel adjoint)
Given NA→B , the channel adjoint N †B→A satisfies
〈Y ,N (X )〉 = 〈N †(Y ),X 〉 ∀X ∈ HA,Y ∈ HB
Every channel can be replicated by an isometry acting on a larger input.
Definition (Isometric extension)
Given a channel NA→B , an isometric extensionU : HA ⊗HE → HB ⊗HE ′ of N satisfies
TrE ′(U(ρ⊗ |0〉〈0|E )U†) = NA→B(ρ) (4)
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 18 / 31
The second P: Petz recovery map QIT crash course
Assumptions
Assume we have the following quantum circuits to start with:
1 Block-encodings of two statesOne can efficiently block-encode density matrices (proof omitted).
The implicit state σA (Uσ =
[σA ·· ·
])
The state N (σA) (UN (σ) =
[N (σA) ·· ·
]).
2 UNE ′A→EB , a unitary extension of the forward channel NSetting: we have characterized the noise and can simulate it usingquantum gates.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 19 / 31
The second P: Petz recovery map QIT crash course
Assumptions
Assume we have the following quantum circuits to start with:
1 Block-encodings of two statesOne can efficiently block-encode density matrices (proof omitted).
The implicit state σA (Uσ =
[σA ·· ·
])
The state N (σA) (UN (σ) =
[N (σA) ·· ·
]).
2 UNE ′A→EB , a unitary extension of the forward channel NSetting: we have characterized the noise and can simulate it usingquantum gates.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 19 / 31
The second P: Petz recovery map QIT crash course
Assumptions
Assume we have the following quantum circuits to start with:
1 Block-encodings of two statesOne can efficiently block-encode density matrices (proof omitted).
The implicit state σA (Uσ =
[σA ·· ·
])
The state N (σA) (UN (σ) =
[N (σA) ·· ·
]).
2 UNE ′A→EB , a unitary extension of the forward channel NSetting: we have characterized the noise and can simulate it usingquantum gates.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 19 / 31
The second P: Petz recovery map QIT crash course
A peek at the Petz recovery map
Given: quantum state σA (implicit ‘input’ to channel ∼ pX ), quantumchannel NA→B , Petz map is:
Pσ,NB→A(ωB) := σ1/2A N †
(N (σA)−1/2ωBN (σA)−1/2
)σ
1/2A ,
Composition of 3 CP maps (overall trace-preserving):
(·)→ [N (σA)]−1/2 (·) [N (σA)]−1/2
(·)→ N †(·),(·)→ σ
1/2A (·)σ1/2
A .
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 20 / 31
The second P: Petz recovery map QIT crash course
A peek at the Petz recovery map
Given: quantum state σA (implicit ‘input’ to channel ∼ pX ), quantumchannel NA→B , Petz map is:
Pσ,NB→A(ωB) := σ1/2A N †
(N (σA)−1/2ωBN (σA)−1/2
)σ
1/2A ,
Composition of 3 CP maps (overall trace-preserving):
(·)→ [N (σA)]−1/2 (·) [N (σA)]−1/2
(·)→ N †(·),(·)→ σ
1/2A (·)σ1/2
A .
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 20 / 31
The second P: Petz recovery map Algorithm
Re-writing the channel adjoint
Second step of map: (·)→ N †(·)Can write adjoint N † in terms of unitary extension UN :
N †(ωB) = 〈0|E ′UN † (IE ⊗ ωB)UN |0〉E ′
Problem: IE is not a quantum state.Solution: act on maximally-entangled state 1
dE
∑dE−1i=0 |i〉E |i〉E , whose
density matrix is ∼ identity.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 21 / 31
The second P: Petz recovery map Algorithm
Re-writing the channel adjoint
Second step of map: (·)→ N †(·)Can write adjoint N † in terms of unitary extension UN :
N †(ωB) = 〈0|E ′UN † (IE ⊗ ωB)UN |0〉E ′
Problem: IE is not a quantum state.Solution: act on maximally-entangled state 1
dE
∑dE−1i=0 |i〉E |i〉E , whose
density matrix is ∼ identity.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 21 / 31
The second P: Petz recovery map Algorithm
Overview of algorithm
Implement an isometric extension of the Petz map:
(I) [N (σA)]−1/2 (·) [N (σA)]−1/2
(II) N †(·)
(III) σ1/2A (·)σ1/2
A .
Finally: tracing over environment E implements the map.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 22 / 31
The second P: Petz recovery map Algorithm
Overview of algorithm
Implement an isometric extension of the Petz map:
I)
(I) [N (σA)]−1/2 (·) [N (σA)]−1/2
(II) N †(·)
(III) σ1/2A (·)σ1/2
A .
Finally: tracing over environment E implements the map.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 22 / 31
The second P: Petz recovery map Algorithm
Overview of algorithm
Implement an isometric extension of the Petz map:
II) I)
(I) [N (σA)]−1/2 (·) [N (σA)]−1/2
(II) N †(·)
(III) σ1/2A (·)σ1/2
A .
Finally: tracing over environment E implements the map.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 22 / 31
The second P: Petz recovery map Algorithm
Overview of algorithm
Implement an isometric extension of the Petz map:
II) I)III)
(I) [N (σA)]−1/2 (·) [N (σA)]−1/2
(II) N †(·)
(III) σ1/2A (·)σ1/2
A .
Finally: tracing over environment E implements the map.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 22 / 31
The second P: Petz recovery map Algorithm
Overview of algorithm
Implement an isometric extension of the Petz map:
II) I)III)
(I) [N (σA)]−1/2 (·) [N (σA)]−1/2
(II) N †(·)
(III) σ1/2A (·)σ1/2
A .
Finally: tracing over environment E implements the map.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 22 / 31
The second P: Petz recovery map Algorithm
Theorem
For a forward channel N and an implicit input state σA, we can realize anapproximation P of the associated Petz recovery channel P, such that:∥∥PσA,N − PσA,N∥∥� ≤ ε, (5)
with
O(√
dEκN (σ)
)uses of UNAE ′→BE (∼optimal ) (6)
O(poly(dE , κN (σ), κ(σ))
)uses of UσA and UN (σA) (7)
dE is the dimension of the system E , which is at least the Kraus rank ofthe channel N (·).
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 23 / 31
The second P: Petz recovery map Algorithm
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 ·
· ·
]with O
(κN (σ)
)uses of UN (σ).
Map III: Uσ =
[σA ·· ·
]QSVT−→
[σ
1/2A ·· ·
]with O(κσ) uses of Uσ.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 24 / 31
The second P: Petz recovery map Algorithm
Map II: Channel adjoint N †(·)
1 Tensor in the maximally entangled state ΓEE/dE
2 Perform UN†.
3 Measure the system E ′, accepting if the all-zeros outcome occurs.
4 Ignore the system E (i.e. trace it out).
Easy peasy, no QSVT needed.
Not contiguous with steps 1 and 2! Do these steps after Map III
which applies σ12A.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 25 / 31
The second P: Petz recovery map Algorithm
Map II: Channel adjoint N †(·)
1 Tensor in the maximally entangled state ΓEE/dE
2 Perform UN†.
3 Measure the system E ′, accepting if the all-zeros outcome occurs.
4 Ignore the system E (i.e. trace it out).
Easy peasy, no QSVT needed.
Not contiguous with steps 1 and 2! Do these steps after Map III
which applies σ12A.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 25 / 31
The second P: Petz recovery map Algorithm
Map II: Channel adjoint N †(·)
1 Tensor in the maximally entangled state ΓEE/dE
2 Perform UN†.
3 Measure the system E ′, accepting if the all-zeros outcome occurs.
4 Ignore the system E (i.e. trace it out).
Easy peasy, no QSVT needed.
Not contiguous with steps 1 and 2! Do these steps after Map III
which applies σ12A.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 25 / 31
The second P: Petz recovery map Algorithm
More on measurement
Implementing in sequence the unitaries created through SVT obtains theoverall unitary
W =
[14
√1
dEκN (σ)V ·
· ·
](8)
where V= ideal Petz map, ‖V − V ‖ < O(ε).
This is a probabilistic implementation: Measuring E ′ system,probability psuccess = O( 1
dEκ) of getting |0〉.
Make this deterministic: use Oblivious Amplitude Amplification toboost probability by repeating W O
(1/√psuccess
)times .
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 26 / 31
Application of our algorithms:Pretty-Good Measurements
The last P: Pretty-Good Measurements
Pretty-Good Measurements
Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, weare promised that ρ is in state σx with probability p(x). What POVMmaximizes Pr(correctly identify ρ)?
No optimal strategy when |X | ≥ 3, but ‘pretty-good measurement’does pretty-well on this. [Belavkin’75, Hausladen-Wootters’94]
Special case of the Petz map with σXB =∑
x pX (x)|x〉〈x |X ⊗ σxB , andN the partial trace over X .
Our Petz map algorithm can implement this, with O(√|X |poly(κ)
)uses of unitary preparing σXB .
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 28 / 31
The last P: Pretty-Good Measurements
Pretty-Good Measurements
Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, weare promised that ρ is in state σx with probability p(x). What POVMmaximizes Pr(correctly identify ρ)?
No optimal strategy when |X | ≥ 3, but ‘pretty-good measurement’does pretty-well on this. [Belavkin’75, Hausladen-Wootters’94]
Special case of the Petz map with σXB =∑
x pX (x)|x〉〈x |X ⊗ σxB , andN the partial trace over X .
Our Petz map algorithm can implement this, with O(√|X |poly(κ)
)uses of unitary preparing σXB .
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 28 / 31
The last P: Pretty-Good Measurements
Pretty-Good Measurements
Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, weare promised that ρ is in state σx with probability p(x). What POVMmaximizes Pr(correctly identify ρ)?
No optimal strategy when |X | ≥ 3, but ‘pretty-good measurement’does pretty-well on this. [Belavkin’75, Hausladen-Wootters’94]
Special case of the Petz map with σXB =∑
x pX (x)|x〉〈x |X ⊗ σxB , andN the partial trace over X .
Our Petz map algorithm can implement this, with O(√|X |poly(κ)
)uses of unitary preparing σXB .
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 28 / 31
The last P: Pretty-Good Measurements
Pretty-Good Measurements
Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, weare promised that ρ is in state σx with probability p(x). What POVMmaximizes Pr(correctly identify ρ)?
No optimal strategy when |X | ≥ 3, but ‘pretty-good measurement’does pretty-well on this. [Belavkin’75, Hausladen-Wootters’94]
Special case of the Petz map with σXB =∑
x pX (x)|x〉〈x |X ⊗ σxB , andN the partial trace over X .
Our Petz map algorithm can implement this, with O(√|X |poly(κ)
)uses of unitary preparing σXB .
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 28 / 31
The last P: Pretty-Good Measurements
Faster PGM
Special case: ensemble of r pure states over a uniform distribution{p(j) = 1/r , |φj〉}r−1
j=0 .
Not that uncommon: this setting was considered by [Holevo’79].
Let κ = 1/σmin(∑
j |j〉〈φj |).
Petz map algorithm: O(r2κ3
)uses of Uφ.
Polar decomposition algorithm can handle this, with polynomialspeedup: O(rκ) uses of Uφ.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 29 / 31
The last P: 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: PGM ∼ optimal measurement todistinguish states
i Is the optimal measurement in q. algorithm for dihedral hiddensubgroup problem [Bacon-Childs-vanDam’06]
ii Bounds on sample complexity for Quantum Probably ApproximatelyCorrect (PAC) learning [Arunachalam-de Wolf’16]
iii Optimal for port-based teleportation [Leditzky’20]
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 30 / 31
The last P: 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: PGM ∼ optimal measurement todistinguish states
i Is the optimal measurement in q. algorithm for dihedral hiddensubgroup problem [Bacon-Childs-vanDam’06]
ii Bounds on sample complexity for Quantum Probably ApproximatelyCorrect (PAC) learning [Arunachalam-de Wolf’16]
iii Optimal for port-based teleportation [Leditzky’20]
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 30 / 31
The last P: Pretty-Good Measurements
Takeaways!
The quantum singular value transform allows us to be systematic, rigorousand fast.
Our algorithm for Polar decomposition(arxiv:20??.?????)
provides new tools for quantum linear algebra;and
speeds up PGM, for pure states and uniformdistribution.
Our algorithm for Petz recovery maps andgeneral-purpose PGM (arxiv:2006.16924)
brings these theoretical tools closer toimplementation; and
is almost optimal in gate complexity.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 31 / 31
The last P: Pretty-Good Measurements
Takeaways!
The quantum singular value transform allows us to be systematic, rigorousand fast.
Our algorithm for Polar decomposition(arxiv:20??.?????)
provides new tools for quantum linear algebra;and
speeds up PGM, for pure states and uniformdistribution.
Our algorithm for Petz recovery maps andgeneral-purpose PGM (arxiv:2006.16924)
brings these theoretical tools closer toimplementation; and
is almost optimal in gate complexity.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 31 / 31
The last P: Pretty-Good Measurements
Takeaways!
The quantum singular value transform allows us to be systematic, rigorousand fast.
Our algorithm for Polar decomposition(arxiv:20??.?????)
provides new tools for quantum linear algebra;and
speeds up PGM, for pure states and uniformdistribution.
Our algorithm for Petz recovery maps andgeneral-purpose PGM (arxiv:2006.16924)
brings these theoretical tools closer toimplementation; and
is almost optimal in gate complexity.
Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 31 / 31