Unitary designs in quantum information science

第35回代数的組み合わせ論シンポジウムでの発表資料

Transcript

1. Unitary designs in quantum information science Yoshifumi Nakata [1,2] In collaboration with C. Hirche [1], M. Koashi [2], and A. Winter [1,3] [1] Autonomous University of Barcelona [2] The University of Tokyo [3] ICREA
2. Outline 1. Unitary designs in Quantum Information in brief 2. Local permutation check problem 1. General question 2. Solution to a special case 3. Further combinatorial questions related to the special case 3. Summary 2/17
3. What is Quantum Information? Quantum information processing Information science Quantum mechanics!! Information processing is done by controlling physical properties such as electric current etc… To explore the ultimate information processing, take the fundamentalrules of Physics into account!! 1 bit → 1 qubit (quantum bit) 3/17
4. Quantum Information now ▪ Huge investment by many companies and governments. ▪ ▪ It will take at least a decade to realize a useful quantum computer. ▪ Do not expect too much too early! 1 qubit < 50 qubits By google By google ≈ −273℃ ≈ 1 4/17
5. How to describe “quantum system” ▪ A “system” is a complex Hilbert space. ▪ 1 “qubit” = 2-dim. complex Hilbert space ℂ2. ▪ qubits = (ℂ2)⨂. ▪ A basic “operation” is a unitary matrix U. ▪ For 1 qubit, U ∈ (2). ▪ For qubits, U ∈ (2N). In principle, all of above are allowed to exist in this world. However, we can do a little in experiment. 1 qubit By google 5/17
6. QI is all about tensor product! 1 qubit By google ▪ In experiments, we can realize the following unitaries. ▪ On an -qubit system (ℂ2)⨂, • (1,2) ⨂ 2 (3)⨂ 2 (4)⨂…. ⨂2 −1 ⨂2 , where (1,2) ∈ (4). • 2 (1)⨂ 2 (2)⨂(3,4) ⨂…. ⨂ 2 (−1)⨂ 2 (), where (3,4) ∈ (4). • 2 (1)⨂ 2 (2)⨂ 2 (3)⨂(4,−1)⨂…. ⨂ 2 (), where (1,3) ∈ (4). • etc… ∈ () For a given U ∈ (2N), decomposing U into a product of local unitaries is important. It is always possible, but how short can the product be? For a given U ∈ (2N), what is where each (,) is a local unitary? “Local unitaries” 6/17
7. Unitary designs in QI ▪ Unitary -designs on (2N) are extremely useful for many purposes!! ▪ Randomness is always useful in information processing. ▪ Especially, = 2, 4 are important. Unitary design on (2N) by a product of local unitaries?? ▪ What is the minimum of ? ▪ Clifford group for = 2, 3. [DiVincenzo et al 2002, Zhu 2017] ▪ Approximate unitary -designs ▪ Harrow and Low, 2009. ▪ Brandao, Harrow, and Horodecki, 2016. ▪ Nakata, Hirche, Koashi, and Winter, 2017. In this specific construction, we met interesting(?) combinatorial problems. 7/17
8. Good bye, quantum information… Let’s get into combinatorics!!
9. Local permutation check 1 = 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 9/17
10. Local permutation check 2 = 1 , 2 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 an 2 -local perm. of ′. • is NOT a row perm. of ′. 10/17
11. Local permutation check 3 11/17
12. Special case (most important to us) ▪ Can be proven by mapping the problem to a geometrical one. ▪ The key lemma is about hypercubes and hyperplanes!! 12/17
13. The Lemma 1 is also interesting? Generalization (probably no application to QIP, but interesting.) [N. Melo and A. Winter, arXiv:1712.01763] 13/17
14. Covering a hypercube by hyperplanes [N. Melo and A. Winter, arXiv:1712.01763] = 3 & = 2 An observation: 14/17
15. 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 15/17
16. Summary ▪ Unitary designs are useful in QIP!!! ▪ Decomposition into local unitaries are important. ▪ Local permutation check problem in our constructions of designs. [N. Melo and A. Winter, arXiv:1712.01763] 16/17
17. Thank you 1. YN, C. Hirche, M. Koashi, and A. Winter, Phys. Rev. X, 7, 021006 (2017). 2. N. Melo and A. Winter, arXiv:1712.01763
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19. Proof sketch 2 Denote each column vector of K and K’ by and ′ ( ∈ [1, ]), respectively. Since ∙ = (# of 1’s in the th column) and ∙ = (# of 11’s in {,} ), The is not a permutation operator since is not a row permutation of ’. 19/17
20. Proof sketch 1 Denote each column vector of K and K’ by and ′ ( ∈ [1, ]), respectively. Since ∙ = (# of 1’s in the th column) and ∙ = (# of 11’s in {,} ), = 4 & = 5 2 4 2 ∙ 2 = 3 2 ∙ 4 = 2 4 ∙ 4 = 3 In arbitrary order ⟺ 20/17