Unitary designs: constructions and applications

Slides for 2017 International Workshop on Quantum Information, Quantum Computing and Quantum Control.

Transcript

1. UNITARY DESIGNS CONSTRUCTIONS AND APPLICATIONS YOSHIFUMI NAKATA UNIVERSITAT AUTONOMA DE BARCELONA (UAB) THE UNIVERSITY OF TOKYO (UT) WITH C. HIRCHE (UAB), M. KOASHI (UT), AND A. WINTER(UAB) PRX 7, 021006 (2017) (SEE ALSO ARXIV:1609.07021) Slides are available at: https://sites.google.com/view/yoshifuminakata/home
2. OUTLINE 1. Introduction • Why Haar random unitary & unitary designs in quantum information science? • Main results – new quantum circuits for unitary designs – 2. Constructions of designs • Quantum circuits • Combinatorics – local permutation check – • Hamiltonian construction – Sherrington-Kirkpartick spinglass – 3. Conclusion and outlook
3. A HAAR RANDOM UNITARY A Haar random untiary A Haar random unitary is the unique unitarily invariant probability measure H on the unitary group (). (A) H( ) = 1, (B) for any ∈ () and any (Borel) set ⊆ , H = H = H . A uniform distribution of the Haar measure Unitary group
4. A HAAR RANDOM UNITARY IN QIP 1 • A Haar random unitary is the most important tool in quantum information science. In Quantum Information Processing • Quantum communication – Random encoding to achieve the quantum capacity [Hayden et al ‘07] • Quantum computation – To show quantum computational supremacy [Mann & Bremner ‘17] • Experiments of quantum devices – Randomized benchmarking [Knill et.al ‘08] – Calibration of multi-qubit Q.circuits [Boixo ‘16] • Q. encryption, Q. algorithms, etc… In fundamental physics • Disordered many-body systems – Random matrix theory [Metha ‘04] • Foundation of quantum statistical mechanics – Thermalization in isolated systems [Popescu et al ‘07] – Pre-thermalization [Reimann ‘16] • Quantum black holes – Solving the information paradox [Hayden & Preskill ‘07] – Holographic codes [Patawskiet al ‘15] • Quantum chaos – Out-of-time-ordered correlators [Roberts & Yoshida ‘16]
5. A HAAR RANDOM UNITARY IN QIP 2 • A Haar random unitary is the most important tool in quantum information science. • An -qubit system is described by a Hilbert space ℋ = (ℂ2)⨂. – In QIP, we need Haar random unitaries on (2)…. • Can we generate Haar random unitaries on such a large dimension? – In QIP, we use quantum circuits. The basic unit is a qubit. = a unit vector in ℂ2. Exponentially large in
6. • Quantum circuit – a sequence of quantum gates- – Quantum gate on qubits is a unitary ∈ 2 , where is independent of . A HAAR RANDOM UNITARY IN QIP 3 2nd qubit 3rd qubit 1 ∈ (22) th qubit ( − 1)th qubit Time 1st qubit 2 ∈ (23) 3 4 5 1 = 1 ⨂ 2−2×2−2 ∈ (2) 2 = 2×2 ⨂2 ⨂ 2−4×2−4 ∈ (2) circuit = … 2 1 ∈ (2) (: # of gates in the circuit) When = poly(), the circuit is efficient. ℋ = (ℂ2)⨂
7. A HAAR RANDOM UNITARY IN QIP 4 • How many gates are needed to implement Haar random unitaries on N qubits? – At least ( ) quantum gates, which is extremely inefficient!! – We can never use a Haar random unitary in practice… • In QIP, a random unitary approximating Haar random unitaries is enough ☺ – A unitary -design comes into the play in QIP! 2nd qubit 3rd qubit th qubit ( − 1)th qubit 1st qubit 3 4 5 circuit = … 2 1 ∈ (2) (: # of gates in the circuit)
8. THE END OF HAAR IS THE BEGINNING OF DESIGN • Can we generate approximate unitary -designs on qubits efficiently by Q. circuits? − Especially when is large and is small (e.g. = 2,4). A distribution of the Haar measure Unitary group A distribution of a unitary design Unitary group ・ ・ ・ ・ ・ ・ ・ ・ ・・ To simulate only low order moments : one unitary Enough for quantum information processing! Approximate unitary design An -approximate unitary -design is a probability measure , that simulates up to the th order statistical moments of the Haar measure within an error .
9. WHAT ARE KNOWN? • Many constructions of unitary 2-designs [DiVincenzo et al ‘01, Cleve et al ’15 etc] – Due to the fact that the Clifford group forms a unitary 2-design. • Only two constructions were known for -designs ( ≥ 3) on qubits Harrow and Low '09 Brandao et al '13 Our result Quantum circuit Q. Fourier Transformation + Toffoli-type gates Local random circuits Hadamard gates + random diagonal gates Methods Markov chain Gap problem of many-body Hamiltonian Combinatorics # of gates (33) [Brodsky & Hoory ’13] (102) Θ(2) Works for = (/ log ) ≤ poly() = ( )
10. WHAT ARE KNOWN? Harrow and Low '09 Brandao et al '13 Our result Quantum circuit Q. Fourier Transformation + Toffoli-type gates Local random circuits Hadamard gates + random diagonal gates Methods Markov chain Gap problem of many-body Hamiltonian Combinatorics # of gates (33) [Brodsky & Hoory ’13] (102) Θ(2) Works for = (/ log ) = (poly ) = ( ) • Many constructions of unitary 2-designs [DiVincenzo et al ‘01, Cleve et al ’15 etc] – Due to the fact that the Clifford group forms a unitary 2-design. • Only two constructions were known for -designs ( ≥ 3) on qubits
11. OUR MAIN RESULTS Harrow and Low '09 Brandao et al '13 Our result ‘16 Quantum circuit Q. Fourier Transformation + Toffoli-type gates Local random circuits Hadamard gates + random diagonal gates Methods Markov chain Gap problem of many-body Hamiltonian Combinatorics # of gates (33) [Brodsky & Hoory ’13] (102) Θ(2) Works for = (/ log ) = (poly ) = ( ) • Many constructions of unitary 2-designs [DiVincenzo et al ‘01, Cleve et al ’15 etc] – Due to the fact that the Clifford group forms a unitary 2-design. • Only two constructions were known for -designs ( ≥ 3) on qubits
12. 2. NEW CONSTRUCTIONS OF DESIGNS
13. EFFICIENT IMPLEMENTATION OF A DESIGN 1 Main theorem (informal) ∃quantum circuits with Θ(2) gates generating a unitary -design. • Main Theorem = Theorem 1 + Theorem 2 Theorem 1 (informal, Proof skipped) Let and be bases in ℋ satisfying a Fourier-type condition. Define D and D by = diag 1, … , , and = diag (1, … , ), where all phases are random. Then, repeating and D O() times becomes a unitary t-design. Unitary -design 1 2 1 2 However, no efficient quantum circuit for (as = 2, too many phases to be randomized).
14. EFFICIENT IMPLEMENTATION OF A DESIGN 2 Main theorem (informal) ∃quantum circuits with Θ(2) gates generating a unitary -design. • Main Theorem = Theorem 1 + Theorem 2 Theorem 2 (informal) Let be the Pauli-Z basis. Then, can be simulated up to the th order by random diagonal-circuits (RDC()) with Θ(2) gates. Unitary -design 1 2 1 2
15. EFFICIENT IMPLEMENTATION OF A DESIGN 3 Main theorem (informal) ∃quantum circuits with Θ(2) gates generating a unitary -design. • Main Theorem = Theorem 1 + Theorem 2 RDC(S) RDC(S) 1 2 Unitary -design Remark 1: When = Pauli-Z basis, = the Pauli-X basis: = ⨂⨂. Theorem 2 (informal) Let be the Pauli-Z basis. Then, can be simulated up to the th order by random diagonal-circuits (RDC()) with Θ(2) gates.
16. EFFICIENT IMPLEMENTATION OF A DESIGN 4 Main theorem (informal) ∃quantum circuits with Θ(2) gates generating a unitary -design. • Main Theorem = Theorem 1 + Theorem 2 RDC(ℑ) RDC(ℑ) RDC(ℑ) RDC(ℑ) Unitary -design Repeating RDC() and the Hadamard gates is a unitary t-design! Remark 1: When = Pauli-Z basis, = the Pauli-X basis: = ⨂⨂. Theorem 2 (informal) Let be the Pauli-Z basis. Then, can be simulated up to the th order by random diagonal-circuits (RDC()) with Θ(2) gates.
17. Random diagonal circuits Let ℓ ⊆ 1, … , be a subset with a constant # of elements, and be {1 , 2 , … , }. A RDC() is a quantum circuit as follows: RANDOM DIAGONAL CIRCUITS 1 Diagonal in the Pauli-Z basis RDC(S) 1 2 3 4 5 diag 1, 2, … … … All phases are randomly & independently chosen from [0, 2). diag 1, 2, … … … E.g.) S={1 , 2 , 3 , 4 }, where 1 = 1,2 , 2 = 2,4,5 , 3 = 3,5 , and 4 = 1,3,5 .
18. RANDOM DIAGONAL CIRCUITS 2 • For what S, does the RDC() simulate = diag (1, … , 2 ) to the th order? – Equivalent to the local permutation check problem. Diagonal in the Pauli-Z basis RDC(S) ≈ Up to the th order moments diag (, … , )
19. LOCAL PERMUTATION CHECK • A pair of × matrices (, ’) with each entry {0, 1}. • For a given S={1 , 2 , … , }, (1): the rows of K are NOT permutation of the rows of K’. (2): for every , the rows of are the permutation of ′ . 0 1 1 0 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 = ′ = = 3 & = 5 Λ S ≔ {(, ′): (, ′) satisfies the conditions (1) and (2). }
20. LOCAL PERMUTATION CHECK • A pair of × matrices (, ’) with each entry {0, 1}. • For a given S={1 , 2 , … , }, Λ S ≔ {(, ′): (, ′) satisfies the conditions (1) and (2). } (1): the rows of K are NOT permutation of the rows of K’. (2): for every , the rows of are the permutation of ′ . 0 1 1 0 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 = ′ = = 3 & = 5 S = 1 , 2 1 = {1,2}, 2 = {2,4,5} 1 ′1 2 ′2 (, ′) ∈ Λ S
21. LOCAL PERMUTATION CHECK • A pair of × matrices (, ’) with each entry {0, 1}. • For a given S={1 , 2 , … , }, Λ S ≔ {(, ′): (, ′) satisfies the conditions (1) and (2). } (1): the rows of K are NOT permutation of the rows of K’. (2): for every , the rows of are the permutation of ′ . 0 1 1 0 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 = ′ = = 3 & = 5 S = 1 , 2 1 = {1,2}, 2 = {2,4,5} 1 ′1 2 ′2 (, ′) ∈ Λ S Local permutation check For a given S = {1 , 2 , … , }, count the # of (K, K′) in Λ S .
22. LPC VS UNITARY DESIGN 1 Local permutation check For a given S = {1 , 2 , … , }, count the # of (K, K′) in Λ S ( = |Λ S | ). • For what S, does the RDC() simulate = diag (1, … , 2 ) to the th order? – Equivalent to the local permutation check problem. Diagonal in the Pauli-Z basis RDC(S) ≈ Up to the th order moments diag (, … , )
23. LPC VS UNITARY DESIGN 2 Local permutation check For a given S = {1 , 2 , … , }, count the # of (K, K′) in Λ S ( = |Λ S | ). Lemma: For a given S = {1 , … , }, if |Λ S | = O(2 −1 ), then RDC() simulates = diag (1, … , ) up to the th order. Find = {1 , 2 , … , } such that 1. | | = O( − ) 2. (# of gates in the circuit) is as small as possible. • For what S, does the RDC() simulate = diag (1, … , 2 ) to the th order? – Equivalent to the local permutation check problem.
24. • General solutions are not known yet……. • Luckily, we found a good choice: Sp = , : ≠ , , ∈ {1, … , } . – The most non-trivial in the whole analysis. LPC VS UNITARY DESIGN 3 Lemma: |Λ(p)| ≤ 2 t−1 N+2t2 , and Sp = = Θ(2). Find = {1 , 2 , … , } such that 1. | | = O( − ) 2. (# of gates in the circuit) is as small as possible. Theorem 2 The can be simulated up to the th order by RDC(p) with Θ(2) gates.
25. ALL TOGETHER –MAIN THEOREM - Theorem 1 (informal, Proof skipped) If and satisfy a Fourier-type condition, repeating and D O() times becomes a unitary t-design. Unitary -design 1 2 1 2 Theorem 2 The can be simulated up to the th order by RDC(p) with Θ(2) gates.
26. ALL TOGETHER –MAIN THEOREM - Theorem 2 The can be simulated up to the th order by RDC(p) with Θ(2) gates. RDC(Sp ) Unitary -design RDC(Sp ) RDC(Sp ) RDC(Sp ) Theorem 1 (informal, Proof skipped) If and satisfy a Fourier-type condition, repeating and D O() times becomes a unitary t-design.
27. Q. CIRCUITS FOR DESIGNS –MAIN THEOREM- diag 1, ( ∈ {0, 2 +1 , … , 2 +1 }) diag(1, 1,1, ℓ ) (ℓ ∈ {0, 2 /2 +1 , 4 /2 +1 , … , 2 /2 +1 }). 1 1 2 1 3 12 13 1 23 2 Repeat (2 + 1) times 1 1 Main theorem Suppose that = ( ). Then, the following quantum circuit forms an -approximate unitary -design, where we use Θ(2 + log2 (1/))) two-qubit gates.
28. HAMILTONIANS FOR DESIGNS Corollary Let () be a time-dependent Sherrington-Kirkpatrick spin-glass Hamiltonian, Then, the time evolution operators generated by () forms a unitary -design after th ≈ 2. Main theorem Suppose that = ( ). Then, the following quantum circuit forms an -approximate unitary -design, where we use Θ(2 + log2 (1/))) two-qubit gates.
29. 3. CONCLUSION AND OUTLOOK
30. CONCLUSION 1. More efficient quantum circuits of approximate unitary -designs. 2. A local permutation check problem is proposed. – Found one specific solution. 3. Point out that the Sherrington-Kirkpatrick spin-glass Hamiltonian ↔ unitary design. – Toy model for OTOC and quantum duality? Harrow and Low ’09 Brandao et al ‘13 Our result ‘16 Methods Markov chain Many-body Hamiltonian Combinatorics Length of the circuit (33) [Brodsky & Hoory ’13] (102) Θ(2) Works for = (/ log ) = (poly ) = ( )
31. OUTLOOK • General solution to the local permutation check? – May result in more efficient quantum circuits. • Lower bounds on the # of quantum gates to generate unitary designs? – Trivial bound is (), possible to show (log). • Efficient quantum circuits for exact ( = 0) unitary -designs on qubits? − None is known for ≥ 4. − From the Carathéodory's theorem, there exist exact designs. THANK YOU FOR YOUR ATTENTION! PRX 7, 021006 (2017) (SEE ALSO ARXIV:1609.07021) Slides are available at: https://sites.google.com/view/yoshifuminakata/home