Decoupling with random diagonal-unitaries Yoshifumi Nakata Universitat Autonoma de Barcelona & The University of Tokyo arXiv:1502.07514 & arXiv:1509.05155 Joint work with C. Hirche, C. Morgan, and A. Winter 2 September 2016 Asian Quantum Information Processing

Outline Introduction Decoupling Haar random unitaries and unitary t-designs Decoupling with random diagonal-unitaries (RDU) Basic idea Decoupling with RDUs Efficient implementations of RDUs By quantum circuits By Hamiltonian dynamics Conclusion and open questions

What is Decoupling? Alice Bob Reference 𝑁 𝐴 qubits CPTP map Reference Choose a good unitary 𝑢 𝐴 such that where 𝜏 𝐵 = tr 𝐴 𝜏 𝐴𝐵 and 𝜌 𝑅 = tr 𝐴 𝜌 𝐴𝑅 . Choi-Jamiolkowski state of the CPTP map.

When and How can we achieve decoupling? What is Decoupling? Alice Bob 𝑁 𝐴 qubits CPTP map Reference When and How can we achieve decoupling? Decoupling is NOT always possible: E.g. 𝜌 𝐴𝑅 is entangled, the CPTP map is an identity map.

Why do we care? Alice Bob Reference CPTP map 𝑁 𝐴 qubits CPTP map Reference When and How can we achieve decoupling? Decoupling provides a free decoder in Q. capacity theorems. plays key roles in fundamental physics: Black hole information science & Thermalisation phenomena Microscopic dynamics of decoupling is NOT fully understood. One of the questions in this talk.

What is Decoupling? Alice Bob Reference CPTP map 𝑁 𝐴 qubits CPTP map Reference When and How can we achieve decoupling? How? Choose u A to be a Haar random unitary. When? A unitary drawn uniformly at random w.r.t. the Haar measure.

A 𝜹-approximate unitary 2-design! What is Decoupling? A 𝛿-approximate unitary 2-design Alice Bob 𝑁 𝐴 qubits CPTP map Reference When and How can we achieve decoupling? How? Choose u A to be a Haar random unitary. When? When the decoupling rate is small. A 𝜹-approximate unitary 2-design! A random unitary that simulates the 2nd order statistical moments (e.g. Variance) of a Haar random unitary within an error 𝜹. [Duplis et.al ’09]

Decoupling Theorem For a 𝛿-approximate unitary 2-design 𝑢 𝐴 Decoupling with approximate designs [Szehr et.al ’11] For a 𝛿-approximate unitary 2-design 𝑢 𝐴 where and 𝑑 𝐴 = 2 𝑁 𝐴 . If 𝛿≤1/ 𝑑 𝐴 4 , decoupling is achieved at the rate of Λ Haar . Natural questions: Is this result tight? Do we really need 𝜹=𝟏/ 𝒅 𝑨 𝟒 to achieve decoupling? (Larger 𝛿 is easier to implement.)

Questions in this talk Questions Can we achieve decoupling by δ-approximate 2-designs where 𝜹≥𝟏/ 𝒅 𝑨 𝟒 ? Physically natural realisation of decoupling? In this talk We provide a new construction of decoupling, based on random diagonal-unitaries. We show that 𝛿≤1/ 𝑑 𝐴 4 is NOT necessary. decoupling can be realised by periodically changing spin-glass-type Hamiltonians.

Outline Introduction Decoupling Haar random unitaries and unitary t-designs Decoupling with random diagonal-unitaries (RDU) Basic idea Decoupling with RDUs Efficient implementations of RDUs By quantum circuits By Hamiltonian dynamics Conclusion and open questions

Basic idea 𝐷 𝑍 = diag 𝑍 𝑒 𝑖 𝜃 1 , 𝑒 𝑖 𝜃 2 ,………, 𝑒 𝑖 𝜃 𝑑 To use random unitaries diagonal in the Z- and X-basis All of θ k and φ k are randomly and independently chosen from [0, 2π). Repeat 𝑫 𝒁 and 𝑫 𝑿 many times: Each 𝐃 𝐢 𝐙 and 𝐃 𝐢 𝐗 are independent and random. 𝐷 𝑍 = diag 𝑍 𝑒 𝑖 𝜃 1 , 𝑒 𝑖 𝜃 2 ,………, 𝑒 𝑖 𝜃 𝑑 𝐷 𝑋 = diag 𝑋 𝑒 𝑖 𝜑 1 , 𝑒 𝑖 𝜑 2 ,………, 𝑒 𝑖 𝜑 𝑑 CPTP map

Every time, 𝑫 𝒁 and 𝑫 𝑿 are chosen independently at random. Basic idea Why do we expect it works? The 𝐷 ℓ is a unitary t-design, if ℓ≥𝑶(𝒕) (in preparation). 𝝋 𝜽 𝝑 U(d) Every time, 𝑫 𝒁 and 𝑫 𝑿 are chosen independently at random.

Decoupling with 𝐷[ℓ] CPTP map 𝑁 𝐴 qubits CPTP map Decoupling Theorem [YN, CH, CM, and AW: arXiv:1509.05155] For 𝐷[ℓ], the following holds: ℓ=𝟑 is sufficient to achieve decoupling.

Decoupling with 𝐷[ℓ] 𝑁 𝐴 qubits Decoupling Theorem’ [YN, CH, CM, and AW: arXiv:1509.05155] When the CPTP map is the partial trace, for ℓ≥𝟐, ℓ=𝟑 is sufficient to achieve decoupling. In the most important cases of the partial traces, ℓ=𝟐 suffices.

Proof Sketch Alice Reference Ãlice Reference CPTP map M.E.S 𝑁 𝐴 qubits M.E.S CPTP map Reference Ãlice Reference An upper bound of ? It’s obtained from the operator In terms of , Use the key lemma for (arXiv: 1502.07514). Still complicated, but durable 

Decoupling and 2-designs Decoupling Theorem [YN, CH, CM, and AW: arXiv:1509.05155] The 𝐷 ℓ for ℓ≥𝟑 (ℓ≥𝟐) achieves decoupling at the rate of O(Λ Haar ) for any CPTP (partial traces) map. How good approximate 2-designs are they? the 𝐷[3] is 𝑶(𝟏/ 𝒅 𝑨 𝟑 )≲𝜹≲𝑶(𝟏/ 𝒅 𝑨 ). the 𝐷[2] is 𝑶(𝟏/ 𝒅 𝑨 𝟐 )≲𝜹≲𝑶(𝟏). Theorem [YN, CH, CM, and AW, arXiv:1502.07514] The 𝐷[ℓ] on 𝑵 qubits is a 𝛿-approximate unitary 2-design, where 1/ 2 ℓ𝑁 ≤𝛿≤2/ 2 (ℓ−2)𝑁 . An 𝑂(1/ 𝑑 𝐴 4 )-app. 2-design is NOT necessary for decoupling.

Are these results optimal? CPTP map Decoupling! Converse statement (weak) Conjecture: ℓ≥𝟐 (ℓ≥𝟏) suffices for any CPTP (partial traces) map. CPTP map Decoupling!

Outline Introduction Decoupling Haar random unitaries and unitary t-designs Decoupling with random diagonal-unitaries (RDU) Basic idea Decoupling with RDUs Efficient implementations of RDUs By quantum circuits By Hamiltonian dynamics Conclusion and open questions

Implementation of 𝐷 𝑍 𝐷 𝑍 = diag 𝑍 𝑒 𝑖 𝜃 1 , 𝑒 𝑖 𝜃 2 ,………, 𝑒 𝑖 𝜃 𝑑 Both 𝐷 𝑍 and 𝐷 𝑋 use exponentially many random numbers. 𝑑=2 𝑁 in N-qubit systems. No way to implement efficiently... A way of approximating lower order properties of 𝐷 𝑍 is known [YN, Koashi, Murao ’14].

Quantum circuits 𝐷 𝑍 𝑫[ℓ]= ×𝟐ℓ dia g 𝑧 (1, 1,1, 𝑒 𝑖 𝜃 𝑘ℓ ) Up to the 2nd order 𝜑 𝑁−1 𝜑 𝑁 𝜃 12 𝜃 13 𝜃 1,𝑁−1 𝜃 1𝑁 𝜑 1 𝜑 2 𝜑 3 𝜃 23 𝜃 𝑁−1,𝑁 𝐻 ×𝟐ℓ Up to the 2nd order 𝑫[ℓ]= dia g 𝑍 1, 𝑒 𝑖𝜑 𝑘 ( 𝜑 𝑘 ∈{0, 2𝜋 3 , 4𝜋 3 }) dia g 𝑧 (1, 1,1, 𝑒 𝑖 𝜃 𝑘ℓ ) ( 𝜃 𝑘ℓ ∈{0,𝜋}). All gates in the 𝐷 𝑍 part are commuting. can be applied simultaneously = Short implementation.

Hamiltonian dynamics Time Decoupled!! Hamiltonian 𝟎 𝝅 𝟐𝝅 𝟑𝝅 4𝝅 5𝝅 6𝝅 7𝝅 Time Decoupled!! Decoupling by spin-glass type Hamiltonians. The time necessary to achieve decoupling is independent of the number of particles. All-to-all interactions are maybe feasible in cavity QED or in semiquantal spin gasses.

Conclusion We have presented a new construction of decoupling based on random X- and Z-diagonal unitaries. We have shown that ℓ=𝟑 (ℓ=𝟐) suffices to achieve decoupling for any CPTP (partial traces) map, implying precise designs are not needed. Decoupling can be achieved by simple quantum circuits. Decoupling can be realised by periodically changing spin- glass-type Hamiltonians. CPTP map

Thank you for your attention! Open questions Are 2-designs really needed to achieve decoupling at the rate of Λ Haar ? Many believe NO. Nobody knows how to show that. 1-designs cannot, what is 1.5-designs? In decoupling with 𝐷 ℓ , how many repetitions are needed? Conjecture: ℓ≥𝟐 suffices for any CPTP map. Is it possible to achieve decoupling by time-independent Hamiltonians? In closed systems, Hamiltonians should be time-indep… Thank you for your attention!