Quantum Information Science and Unitary Designs

2018年10月4日　東北大学情報科学　研究科重点プロジェクト　講演会資料

Transcript

1. Quantum information science and Unitary designs 中田芳史 東京大学 光量子科学研究センター 特任研究員
2. Self-introduction 中田芳史 東京大学工学系 光量子科学研究センター: 特任研究員 ❑ 経歴： ❑ 専門：量子情報科学 ➢ 2006 - 2008: 東京大学 修士課程（理学） ➢ 2008 - 2010: 青年海外協力隊 エチオピア ➢ 2008 - 2013: 東京大学 博士課程（理学） ➢ 2013 - 2015: Leibniz University Hannover (Germany) ➢ 2015 - 2017: Autonomous University of Barcelona (Spain) ➢ 2017 - : 東京大学（特任研究員） ➢ 2018 - : 京都大学基研（特定助教） 物理と情報科学の学際分野 最近は「ユニタリ・デザイン」に 関連した研究 2/52
3. Self-introduction 中田芳史 東京大学工学系 光量子科学研究センター: 特任研究員 ❑ 今、量子情報でユニタリ・デザインが熱い！！ ➢ 量子情報におけるユニタリ・デザインの重要性 ➢ 今後、数学と量子情報の研究交流につなげたい 参考文献：上記シンポジウムの報告書（未アップロード） https://hnozaki.jimdo.com/proceedings-symp-alg-comb/ 中田のHP (Googleで「google site, yoshifumi nakata」で検索！) 3/52
4. 1. Quantum Information 1. Quantum Mechanics (QM) 2. Information processing based on QM 2. Unitary designs in Quantum Information 1. “Efficiency” of unitary matrices 2. Constructing “efficient” unitary designs 3. Combinatorial problems 1. Local permutation check 2. Covering hypercube by hyperplane Outline of this talk 物理 数学 4/52
5. 1. Quantum Information 2. Unitary designs in Quantum Information 3. Combinatorial problems Outline of this talk 2 Quantum Information is Fun ☺ “Efficiency” of unitary is very important. More collaboration desired!! (Math & Quantum Information!) 5/52 Mathematics
6. 1. Quantum Information 1-1. weird quantum mechanics 1-2. Information processing based on QM
7. 1. Quantum Information 量子 情報 • Computation • Communication • Cryptography • etc… Information processing based on quantum mechanics 量子力学 Most fundamental theory in physics
8. Quantum Mechanics (QM) Very basics in quantum theory ❑ Quantum State (量子状態) ❑ Quantum measurement (量子測定） ❑ Quantum dynamics (時間発展) ❑ Quantum State (量子状態) ➢ Describes all properties of particles. E.g.) energy, position, etc… ➢ A Hilbert space ℋ (dim = ) is attached to a particle. ➢ A quantum state is a unit vector ∈ ℋ. Different vector → e.g. different energy, etc… Particle Energy Position Momentum Charge , etc… ℋ Unit vector ∈ ℋ 8/52
9. Quantum Mechanics (QM) Very basics in quantum theory ❑ Quantum State (量子状態): Unit vector ∈ ℋ ❑ Quantum measurement (量子測定) ❑ Quantum dynamics (時間発展) ❑ Quantum measurement (量子測定) ➢ A Hilbert space ℋ (dim = ) is attached to a particle. ➢ A quantum measurement is described by a basis { }=1,…, ∈ ℋ. Measurement { }=1,…, ∈ ℋ Obtain the result with probability = | ∙ |2 Result Particle Energy Position Momentum Charge , etc… ℋ Unit vector ∈ ℋ 9/52
10. Quantum Mechanics (QM) with prob. 1/2 with prob. 1/2 Quantum state Ԧ = 1 2 ( 1 1 ) with prob. 1!! Red ( ) or Blue ( ) ?? Measure Yellow 1 2 ( 1 1 ) or Green 1 2 ( 1 −1 )?? Albert Einstein It’s not a complete theory. 10/52 同じ量子状態(unit vector)でも「量子測定(Basis)の選び方」 によって、結果が変わり、確率的な結果しか得られない。
11. Quantum Mechanics (QM) Very basics in quantum theory ❑ Quantum State (量子状態): Unit vector ∈ ℋ ❑ Quantum measurement (量子測定): Basis { }=1,…, ∈ ℋ ❑ Quantum dynamics (時間発展) ❑ Quantum dynamics (時間発展) ➢ A Hilbert space ℋ (dim = ) is attached to a particle. ➢ A dynamics is a × unitary matrix ∈ () on ℋ. Different energy, position, etc… Unit vector ∈ ℋ Particle Energy Position Momentum Charge , etc… ℋ Unit vector ∈ ℋ Dynamics: 11/52
12. Quantum Information 1 Very basics in quantum theory ❑ Quantum State (量子状態): Unit vector ∈ ℋ ❑ Quantum measurement (量子測定): Basis { }=1,…, ∈ ℋ ❑ Quantum dynamics (時間発展): A × unitary ∈ () on ℋ Quantum Information Science Information processing based on quantum mechanics ❑ Quantum Information Processing (量子情報処理) should be completely different from the current information processing. Bit = {0,1} Qubit = 2-dim Hilbert space { , } Information processing Quantum information processing 12/52
13. Quantum Information 1 Very basics in quantum theory ❑ Quantum State (量子状態): Unit vector ∈ ℋ ❑ Quantum measurement (量子測定): Basis { }=1,…, ∈ ℋ ❑ Quantum dynamics (時間発展): A × unitary ∈ () on ℋ Quantum computation (量子計算) Input Output Computer Bit列: 010110110…. Bit列: 0110100…. Qubit列: ∈ ℋ Qubit列: ∈ ℋ ∈ () Measurement Result Qubit = 2-dim Hilbert space Quantum analog of bit 13/52
14. 1 qubit < 50 qubits By google ≈ −273℃ ≈ 1 Quantum Information 2 ❑Quantum computation is very powerful!! ❑ Do not expect too much too early. Perhaps, 20 years… ➢ A basic unit is qubit, instead of bit. ➢ Huge investment to the field. ℋ Qubit = 2-dim Hilbert space Quantum analog of bit By google Why unitary designs are important in QIP?
15. ❑ Currently, interested in “randomness” in QI. ❑ In Quantum Information Science, 1 qubit < 50 qubits By google ≈ −273℃ ≈ 1 ℋ ➢ Randomness is useful in information processing ➢ Randomness is described by a Haar random unitary. ➢ Approximations suffice in many applications!! Unitary designs By google Randomness in Quantum Infor. Qubit = 2-dim Hilbert space Quantum analog of bit 15/52
16. 2. Unitary designs in Q.I. 2-1. “Efficiency” of unitary matrices 2-2. Constructing “efficient” unitary designs
17. Quantum Information 1 Very basics in quantum theory ❑ Quantum State (量子状態): Unit vector ∈ ℋ ❑ Quantum measurement (量子測定): Basis { }=1,…, ∈ ℋ ❑ Quantum dynamics (時間発展): A × unitary ∈ () on ℋ Quantum computation (量子計算) Input Output Computer Bit列: 010110110…. Bit列: 0110100…. Qubit列: ∈ ℋ Qubit列: ∈ ℋ ∈ () Measurement Result Qubit = 2-dim Hilbert space Quantum analog of bit 17/52
18. ❑ When there are qubits, the total Hilbert space is given by the tensor product! ❑ Depending on the protocol, changes!! By google Tensor product in Q. Info. 18/52
19. By google Remarks: ❑ Some are experimentally “easy”, and other are “difficult”. Local unitaries and Efficiency 2 ➢ Local unitaries!! 19/52
20. By google ❑ The size of the non-trivial part is indep. of . ❑ Local unitaries (esp. 2-local) are essentially only those we can experimentally implement! Local unitaries and Efficiency 3 Theorem: any unitary { }∈ℕ can be written as a product of sufficiently many 2-local unitaries. ➢ In experiment, shorter products are preferable! 20/52
21. ❑ For a given{ }∈ℕ , the product is NOT unique! ❑ Since can be (1010), if the length is (2), we need to use (21010 ) local unitaries…… Local unitaries and Efficiency 4 ➢ If the minimum length is ≤ poly(N), it is efficient. ➢ The minimum length is called an efficiency of the unitary. Efficiency is the most important concept in quantum information!! 21/52
22. 2. Unitary designs in Q.I. 2-1. “Efficiency” of unitary matrices 2-2. Constructing “efficient” unitary designs
23. ❑ The Haar measure is a trivial -design for any . ❑ Unitary designs on qubits, a family { }∈ℕ of designs, is important. Unitary designs 1 ➢ Efficiency is important b/c we want to use in practice! A distribution of the Haar measure Unitary group () A distribution of a unitary design Unitary group () Simulating th order properties : one unitary 23/52
24. Efficient unitary designs 1 A distribution of the Haar measure Unitary group () A distribution of a unitary design Unitary group () Simulating th order properties : one unitary 24/52
25. Efficient unitary designs 1 : efficient unitary A distribution of the Haar measure Unitary group (2) Simulating th order properties Unitary group (2) 25/52 Question: How to construct efficient (as efficient as possible) unitary designs?
26. Efficient unitary designs 2 Remark: ❑ The Haar measure is extremely inefficient!! ❑ Especially, interested in small t, e.g. t=2,4,8….. ❑ ∃ a “bound” on the efficiency (=the number of local unitaries): ➢ Any family { }∈ℕ of unitary -design contains unitaries that are products of local unitaries with length ≈ . Is it possible to achieve a -design with ≈ local unitaries??? Question: How to construct efficient (as efficient as possible) unitary designs? Open problem 26/52
27. Constructing unitary designs For unitary 2-designs: ❑ Relatively, easy problem even for exact ones. ❑ Many efficient constructions are known. Question: How to construct efficient (as efficient as possible) unitary designs? For unitary t-designs (t ≥ 3): ❑ Little is known. ❑ No exact & efficient constructions are known so far. 27/52
28. Efficient unitary 2-designs ❑ Based on this, efficient construction of exact unitary 2-designs is known: Is it possible to achieve 2-design with ≈ local unitaries??? Open problem 28/52
29. ❑ Known efficient constructions of approximate unitary t-designs. Efficient unitary t-designs 1 ➢ HL2009 based on quantum version of expander graph & mixing time of a permutation group. ➢ BHH2012 based on a many-body Hamiltonian problem. ➢ NHKW2017 based on a combinatorial problem called local permutation check. BHH12 Googleによる実験 (超伝導qubit:≈ qubits?) [S. Boixo, Nature Physics, 2018] NHKW17 中国・カナダによる実験 (NMR: 12 qubits) [J. Li, arXiv, 2018] 29/52
30. ❑ Idea is to use “diagonal” unitary matrices in two bases, and : Efficient unitary t-designs 2 Unitary group 30/52
31. ❑ Idea is to use “diagonal” unitary matrices in two bases, and : Efficient unitary t-designs 2 Unitary diagonal in the basis , and consists of ( − 1)/2 local unitaries. Unitary diagonal in the basis , and consists of ( − 1)/2 local unitaries. 31/52
32. ❑ Idea is to use “diagonal” unitary matrices in two bases, and : Efficient unitary t-designs 2 ❑ The # of local unitaries ≈ 2. Product of ( − 1)/2 local unitaries 32/52
33. ❑ Less is known about efficient constructions of unitary t-designs. Efficient unitary t-designs 3 Is it possible to achieve -design with ≈ local unitaries??? 33/52
34. ❑Local unitaries and efficiency of a unitary ❑Efficient unitary designs ➢ Decompose a unitary { } into the products of local unitaries!! ➢ If the length is poly(), the unitary is efficient. Summary of designs in Quantum Information Science By google What is the minimum length of the product? Unitary -design with ≈ local unitaries??? ➢ Especially, small is important. E.g. = 2,4,8… 34/52
35. 3. Combinatorial problems 3-1. Local permutation check 3-2. Covering hypercube by hyperplane
36. = 0 1 1 0 1 0 1 1 1 1 0 0 1 1 1 = 3 & = 5 I = {1,2,5} = 0 1 0 0 1 1 1 1 1 Ω( ) = 1 1 1 0 1 1 0 1 0 Local permutation check 1 36/52
37. 1 = {1,2}, 2 = {2,4,5} = 0 1 1 0 1 0 1 1 1 1 0 0 1 1 1 = 3 & = 5 ′ = 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 • 2 = 1 1 0 1 0 1 1 1 1 & ′2 = 1 1 1 1 0 1 1 1 0 ∴ Ω(2 ) = Ω(′2 ) • Ω(1 ) = Ω(′1 ) • is NOT a row perm. of ′. • is an 1 -local perm. of ′. • is an 2 -local perm. of ′. • is an -local perm. of ′. Local permutation check 2 37/52
38. Local permutation check 3 38/52
39. Various situations can be thought of: ❑ What is , , , 0 ? ❑ When is fixed to be non-zero, what is the minimum such that , , , < ∞? ❑ etc….. Local permutation check 4 Remark: ❑ Mostly, interested in ≪ . ❑ When ≫ , the problem is closely related to “Hadamard design”. 39/52
40. = 2 & = 2 = 3 & = 2 : similar method works. = 4 & = 2 : something happens!! L(t,N,k,0)=? 40/52
41. = 4 & = 2 L(t,N,k,0)=? 41/52
42. = 8 & = 3 L(t,N,k,0)=? 42/52
43. L(t,N,k,0)=? 43/52
44. In particular, (, , , 2 −1 )? ( is only dependent on ) Local permutation check 4 Various situations can be thought of: ❑ What is , , , 0 ? ❑ When is fixed to be non-zero, what is the minimum such that , , , < ∞? ❑ etc….. 44/52
45. 45/52 ❑ What exactly needed? ❑ Why do we care? ∵) related to efficient construction of unitary designs! L(t,N,k, − )=? ➢ Is this optimal?? ➢ Does there ∃constant for which , , , 2 −1 ≈ ? Unitary -design with ≈ local unitaries???
46. 4. Combinatorial problems 4-1. Local permutation check 4-2. Covering hypercube by hyperplane
47. The proof is based on a geometrical approach. [YN, Hirche, Koashi, and Winter, arXiv 1609.07021] [N. Melo and A. Winter, arXiv:1712.01763] Geometrical approach Covering a hypercube by hyperplanes 47/52
48. Covering a hypercube by hyperplanes [N. Melo and A. Winter, arXiv:1712.01763] = 3 & = 2 Observations: 48/52
49. Propositions by Melo an Winter: Conjecture: Observations 1. The largest number is 2. 2. The 2nd largest one is 3 ∙ 2−2. Covering a hypercube by hyperplanes 49/52
50. 5. Summary Time flies…
51. 1. Quantum Information 2. Unitary designs in Quantum Information 3. Combinatorial problems Summary of the talk = information processing based on quantum mechanics. Local permutation check & covering a hypercube by hyperplanes Local unitaries and efficiency are very important. How to construct efficient unitary designs?? 参考文献：上記シンポジウムの報告書（未アップロード） https://hnozaki.jimdo.com/proceedings-symp-alg-comb/ 中田のHP (Googleで「google site, yoshifumi nakata」で検索！) 51/52
52. Thank you