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Los Alamos National Laboratory LA-UR-20-22306 Preparation and Compression of Symmetric Pure Quantum States joint work with Stephan Eidenbenz Andreas B artschi NSEC/CNLS, baertschi@lanl.gov ISTI Seminar (LANL) QMATH (Copenhagen) QIT (ETH

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Los Alamos National Laboratory LA-UR-20-22306

Preparation and Compression of Symmetric Pure Quantum States

joint work with Stephan Eidenbenz

Andreas B¨ artschi NSEC/CNLS, baertschi@lanl.gov ISTI Seminar (LANL) QMATH (Copenhagen) QIT (ETH Z¨ urich) August, 2019

Managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA

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Los Alamos National Laboratory

How efficiently can we prepare symmetric quantum states?

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 2

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Los Alamos National Laboratory

Efficient Symmetric State Preparation

Symmetric n-qubit States, e.g. |ψ = −

√ 2|000+i|011+i|101+i|110 √ 5

= −

√ 2|D3

0+

√ 3i|D3

2

√ 5

Symmetric under permutation of the qubits. All terms with the same Hamming Weight must have the same amplitude: ⇒ Dicke States |Dn

k are equal superpositions of all HW-k strings.

⇒ They form an orthonormal basis for symmetric pure states. Efficient in terms of Circuit Model: Deterministic Scheme, Linear Nearest Neighbor architecture. Small total number of Gates: O(n2) many. Shallow Depth: O(n) steps. Few Ancilla Qubits: None.

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 3

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Los Alamos National Laboratory

Outline

1 Context

How many gates for. . . ? Known upper bounds Results for Symmetric States

2 Dicke States

Dicke State Unitaries Inductive Approach Split & Cyclic Shift Unitaries Combining all Ideas

3 Arbitrary Symmetric Pure States

Preparation of Symmetric Pure States Compression of Symmetric Pure States Conclusion

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 4

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Los Alamos National Laboratory

Context

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 5

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Los Alamos National Laboratory

How many gates for. . . ?

Given n qubits and a unitary U as a 2n × 2n matrix, how many 1- and 2-qubit gates do we need to implement all of U, the first column of U (state preparation), the first K columns of U, |0

            . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .            

|0 |0 |0 |0 exactly? Θ(K · 2n) gates are sufficient and sometimes necessary.[K95, SBM04] approximately?

  • Θ(K · 2n) gates are sufficient and still sometimes necessary!

In fact, the set of states approximately preparable with fewer gates has measure 0.[K95] Which States can we prepare efficiently?

[K95]: Knill, Approximation by Quantum Circuits, 1995 [SBM04]: Shende, Bullock, Markov, Synthesis of Quantum Logic Circuits, 2004

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 6

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Los Alamos National Laboratory

Known upper bounds

A state |ψ = 2n−1

i=0 ψi |i can be prepared with polynomial resources, if e.g.,

there are only polynomially many non-zero amplitudes ψi,[SBM04] ? all ψi are easily computed from i and |ψi|2 ∈ O( 1

2n ).[SS04, GR02]

Our states are “in-between”: States with an intermediate number of non-zero amplitudes, a polynomial number of distinct amplitudes, easily computed amplitudes ψi.

[SBM04]: Shende, Bullock, Markov, Synthesis of Quantum Logic Circuits, 2004 [SS04]: Soklakov, Schack, Efficient state preparation for a register of quantum bits, 2004 [GR02]: Grover, Rudolph, Creating superpositions [..] efficiently integrable probability distributions, 2002

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 7

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Los Alamos National Laboratory

Dicke States

The Dicke state |Dn

k is an equal-weight superposition of all n-qubit computational basis

states with Hamming Weight k, e.g. |D4

2 = 1 √ 6 (|1100 + |1010 + |1001 + |0110 + |0101 + |0011) .

We show how to prepare |Dn

k with

few gates – O(kn) many, low depth – O(n) steps, and no extra ancilla qubits.

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 8

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Results for Symmetric States

Preparation Type Ancillas Circuit Depth Number of Gates [CFGG] Dicke States O(log n) O(n poly logn) O(n poly logn) [C+18] W States O(log n) O(n) [KM04] Symmetric States O(log(n/ε)) O(n poly log(n/ε)) O(n poly log(n/ε)) Our result Dicke States O(n) O(kn) Symmetric States O(n) O(n2)

[CFGG]: Childs, Farhi, Goldstone, Gutmann, Finding cliques by quantum adiabatic evolution, 2000 [C+18]: Cruz et al., Efficient quantum algorithms for GHZ and W states, [..], 2018 [KM04]: Kaye, Mosca, Quantum Networks for Generating Arbitrary Quantum States, 2004

Compression Type Ancillas Circuit Depth Number of Gates [BCH04] Schur Transform O(log(n/ε)) O(n poly log(n/ε)) O(n poly log(n/ε)) [PB09] Symmetric States O(n2) O(n2) Our result Symmetric States O(n poly log n) O(n2)

[BCH04]: Bacon, Chuang, Harrow, Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms, 2004 [PB09]: Plesch, Buzek, Efficient compression of quantum information, 2009

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 9

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Los Alamos National Laboratory

Dicke States

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 10

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Dicke State Unitaries

Definition (Dicke State Unitaries Un,k)

A Dicke State Unitary Un,k is any unitary which for all ℓ ≤ k implements the mapping Un,k : |0⊗n−ℓ |1⊗ℓ = |0..0

  • n−ℓ

1..1

  • ℓ

− → |Dn

ℓ .

“for all ℓ ≤ k”: allows for an Inductive Approach, constructing Un,k inductively over n, allows for Symmetric State Preparation, as Un,n can be used to construct all Dicke States.

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 11

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Los Alamos National Laboratory

Inductive Approach

Write Dicke States inductively as a superposition involving “smaller Dicke states”, grouping terms by the last qubit being |1 or |0: |Dn

ℓ =

n

ℓ

− 1

2

  • x has n qubits

with exactly ℓ 1’s

|x =

  • ℓ

n |Dn−1 ℓ−1 ⊗ |1 +

  • n−ℓ

n |Dn−1 ℓ

⊗ |0 . Assume we know how to design Un−1,k but do not know how to design Un,k: |Dn

ℓ should be prepared by Un,k from |0⊗n−ℓ |1⊗ℓ,

|Dn−1

ℓ−1 can be prepared by Un−1,k from |0⊗n−ℓ |1⊗ℓ−1,

|Dn−1

ℓ

can be prepared by Un−1,k from |0⊗n−ℓ−1 |1⊗ℓ. Observation: The only thing missing is a unitary which implements the mapping |0⊗n−ℓ−1 |0 |1⊗ℓ − →

  • ℓ

n |0⊗n−ℓ−1 |0 |1⊗ℓ−1 |1 +

  • n−ℓ

n |0⊗n−ℓ−1 |1⊗ℓ |0 .

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 12

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Los Alamos National Laboratory

Split & Cyclic Shift Unitaries

Definition (Split & Cyclic Shift Unitaries SCSn,k)

A Split & Cyclic Shift Unitary SCSn,k is any unitary which for all ℓ ∈ 1, . . . , k and k < n implements the mappings SCSn,k : |0⊗k+1 → |0⊗k+1 , SCSn,k : |0⊗k+1−ℓ |1⊗ℓ →

  • ℓ

n |0⊗k+1−ℓ |1⊗ℓ +

  • n−ℓ

n |0⊗k−ℓ |1⊗ℓ |0 ,

SCSn,k : |1⊗k+1 → |1⊗k+1 . SCSn,k unitaries can be built explicitly, we now construct Un,k unitaries inductively.

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 13

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Los Alamos National Laboratory

Review: Gate Overview

Goal: Build Un,k from SCSn,k (acting on the last k + 1 qubits) and Un−1,k (acting on the first n − 1 qubits): SCSn,k : |0..0 → |00..000 |00..001 →

  • 1

n |00..001 +

  • n−1

n |00..010

|00..011 →

  • 2

n |00..011 +

  • n−2

n |00..110

. . . |01..111 →

  • k

n |01..111 +

  • n−k

n

|11..110 |11..111

k+1

→ |11..111 Un−1,k : |0..0 → |Dn−1

  • |0..000..01 → |Dn−1

1

  • |0..000..11 → |Dn−1

2

  • .

. . |0..001..11 → |Dn−1

k−1

|0..0

  • n−1−k

11..11

k

→ |Dn−1

k

  • Andreas B¨

artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 14

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Los Alamos National Laboratory

Inductive Construction of Un,k

The Split & Cyclic Shift unitaries SCSn,k act non-trivially only on the last k + 1 of n qubits, and preceding Un−1,k by SCSn,k we get Un,k.

= = Un,k |0⊗n−k−1 Un−1,k k + 1 n − k: | Split & Cyclic Shift SCS n,k . . . n − 1: | n: | SCS n,k SCS n-1,k SCS k+1,k SCS 3,2 SCS 2,1 . . . . . .

Inductively applying this idea, we construct Un,k by concatenating SCSn,k, SCSn−1,k, . . . , SCSk+1,k as a prefix to Uk,k, construct Uk,k by concatenating SCSk,k−1, SCSk−1,k−2, . . . , SCS2,1.

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 15

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Los Alamos National Laboratory

Explicit Construction of SCSn,k

Recall: SCSn,k : |0⊗k−ℓ |0 |1⊗ℓ−1 |1 →

  • ℓ

n |0⊗k−ℓ |0 |1⊗ℓ−1 |1+

  • n−ℓ

n |0⊗k−ℓ |1 |1⊗ℓ−1 |0

Use blocks of type: (i) |00 → |00 ; |11 → |11 |01 →

  • 1

n |01 +

  • n−1

n |10

(ii) |000 → |000 ; |001 → |001 |010 → |010 ; |111 → |111 |011 →

  • ℓ

n |011 +

  • n−ℓ

n |110

n − 1 Ry(2 cos−1 √ 1

n)

n n − ℓ Ry(2 cos−1 √ ℓ

n)

n − ℓ + 1 . . . n

with (controlled) Y -rotations Ry(2θ) = cos θ − sin θ

sin θ cos θ

  • .

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 16

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Los Alamos National Laboratory

Combining all Ideas

To build Un,k with k ∈ Θ(n): Each of the n − 1 many SCS unitaries described above is constructed with O(n) 3-qubit and O(n) SWAP-gates in a stair-like shape. ⇒ Parallelizes well: O(n2) gates in O(n) steps. ⇒ Even when 2-qubit gates are only allowed between neighbours.

1: |0

√5

6

6

|D6

5 2: |1

√4

6

√4

5

5 3: |1

√3

6

√3

5

√3

4

4 4: |1

√2

6

√2

5

√2

4

√2

3

3 5: |1

√1

6

√1

5

√1

4

√1

3

√1

2

2 6: |1 1

SCS 6,5 SCS 5,4 SCS 4,3 SCS 3,2 SCS 2,1

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 17

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Los Alamos National Laboratory

Combining all Ideas

To build Un,k with k ∈ Θ(n): Each of the n − 1 many SCS unitaries described above is constructed with O(n) 3-qubit and O(n) SWAP-gates in a stair-like shape. ⇒ Parallelizes well: O(n2) gates in O(n) steps. ⇒ Even when 2-qubit gates are only allowed between neighbours. To build Un,k with k ∈ o(n): Make O(n/k) groups of O(k) many SCS unitaries which are constructed with O(k) 3-qubit and O(k) SWAP-gates. ⇒ Parallelizes well: O(kn) gates in O(n) steps. ⇒ Even when 2-qubit gates are only allowed between neighbours.

1-5 1-5 6 6 7 7 8 8 9 9 10

√ 5

15

√ 5

14

√ 5

13

√ 5

12

√ 5

11

10 11

√ 4

15

√ 4

14

√ 4

13

√ 4

12

√ 4

11

11 12

√ 3

15

√ 3

14

√ 3

13

√ 3

12

√ 3

11

12 13

√ 2

15

√ 2

14

√ 2

13

√ 2

12

√ 2

11

13 14

√ 1

15

√ 1

14

√ 1

13

√ 1

12

√ 1

11

14 15 15

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 17

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Los Alamos National Laboratory

Arbitrary Symmetric Pure States

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 18

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Preparation of Symmetric Pure States

Every n-qubit symmetric pure state can be written as a superposition of the n + 1 Dicke states |Dn

0 , |Dn 1 , . . . , |Dn n in the form ℓ eiφℓαℓ |Dn ℓ ,

with magnitudes αℓ ∈ [0, 1], α2

0 + . . . + α2 n = 1, and phases φℓ ∈ [0, 2π), φ0 = 0.

Given the unitary Un,n, it remains to prepare

  • ℓ eiφℓαℓ |0⊗n−ℓ |1⊗ℓ. To this end, we define

magnitudes βℓ such that αℓ = βℓ · ℓ−1

j=0

  • 1 − β2

j ,

and angles ψℓ such that φℓ = ℓ

j=0 ψj.

|0 β4 Rψ5 Un,n

eiφℓαℓ |Dn

ℓ

|0 β3 Rψ4 |0 β2 Rψ3 |0 β1 Rψ2 |0 β0 Rψ1

Using Y -rotations Ry(2 cos−1 βℓ) and phase-shift gates Rψℓ = 1

0 eiψℓ

  • yields:

|0⊗n

β0,...,βn−1

− − − − − − →

  • ℓ αℓ |0⊗n−ℓ |1⊗ℓ

ψ1,...,ψn

− − − − − →

  • ℓ eiφℓαℓ |0⊗n−ℓ |1⊗ℓ

Un,n

− − − − →

  • ℓ eiφℓαℓ |Dn

ℓ .

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 19

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Compression of Symmetric Pure States

Quasilinear-depth quantum compression circuit: Compressing a symmetric n-qubit pure state

ℓ γℓ |Dn ℓ into ⌈log(n + 1)⌉ qubits:

γℓ |Dn

ℓ

U †

n,n

|0 |0 |0 |0 ⌈log(n + 1)⌉ qubits

The inverse unitary U†

n,n maps each Dicke state |Dn ℓ to |0⊗n−ℓ |1⊗ℓ,

which gets mapped to ℓ’s one-hot encoding |0⊗n−ℓ |1 |0⊗ℓ−1, and finally to ℓ’s binary encoding |ℓ (with padded zeroes).

  • ℓ γℓ |Dn

ℓ U†

n,n

− − − − − →

  • ℓ γℓ |0⊗n−ℓ |1⊗ℓ

CNOT stair

− − − − →

  • ℓ γℓ |0⊗n−ℓ |1 |0⊗ℓ−1

encoding change

− − − − − →

  • ℓ γℓ |ℓ .

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 20

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Conclusion

We have shown how to prepare Dicke States |Dn

k – and by extension Symmetric Pure

States – deterministically on circuits with Linear Nearest Neighbor architecture with O(kn) many gates, O(n) depth, and no ancilla qubits. This also gives a quasilinear quantum compression circuit for Symmetric Pure States. Open problems: Can we implement Dicke States in polylogarithmic depth (for small k)? Is there an upper or lower bound for state preparation

  • f states with a small number of distinct non-zero amplitudes?

Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 21