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
Los Alamos National Laboratory How efficiently can we prepare symmetric quantum states? Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 2
Los Alamos National Laboratory Efficient Symmetric State Preparation √ √ √ 2 | D 3 3 i | D 3 Symmetric n -qubit States, e.g. | ψ � = − 2 | 000 � + i | 011 � + i | 101 � + i | 110 � = − 0 � + 2 � √ √ 5 5 Symmetric under permutation of the qubits. All terms with the same Hamming Weight must have the same amplitude: ⇒ Dicke States | D n 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 ( n 2 ) many. Shallow Depth: O ( n ) steps. Few Ancilla Qubits: None. Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 3
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
Los Alamos National Laboratory Context Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 5
Los Alamos National Laboratory How many gates for. . . ? Given n qubits and a unitary U as a 2 n × 2 n matrix, | 0 � . . . . how many 1- and 2-qubit gates do we need . . . . | 0 � to implement . . . . . . . . | 0 � . . . . all of U , . . . . | 0 � the first column of U (state preparation), . . . . . . . . the first K columns of U , | 0 � exactly? Θ( K · 2 n ) gates are sufficient and sometimes necessary. [K95, SBM04] approximately? � Θ( K · 2 n ) 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
Los Alamos National Laboratory Known upper bounds A state | ψ � = � 2 n − 1 i =0 ψ i | i � can be prepared with polynomial resources, if e.g., there are only polynomially many non-zero amplitudes ψ i , [SBM04] ? all ψ i are easily computed from i and | ψ i | 2 ∈ O ( 1 2 n ). [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 ψ i . [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
Los Alamos National Laboratory Dicke States The Dicke state | D n k � is an equal-weight superposition of all n -qubit computational basis states with Hamming Weight k , e.g. | D 4 1 2 � = 6 ( | 1100 � + | 1010 � + | 1001 � + | 0110 � + | 0101 � + | 0011 � ) . √ We show how to prepare | D n 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
Los Alamos National Laboratory Results for Symmetric States Preparation Type Ancillas Circuit Depth Number of Gates [CFGG] Dicke States O (log n ) O ( n poly log n ) O ( n poly log n ) [C+18] W States 0 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 0 O ( n ) O ( kn ) O ( n 2 ) Symmetric States 0 O ( n ) [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 )) O ( n 2 ) O ( n 2 ) [PB09] Symmetric States 0 O ( n 2 ) Our result Symmetric States 0 O ( n poly log n ) [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
Los Alamos National Laboratory Dicke States Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 10
Los Alamos National Laboratory Dicke State Unitaries Definition (Dicke State Unitaries U n , k ) A Dicke State Unitary U n , k is any unitary which for all ℓ ≤ k implements the mapping U n , k : | 0 � ⊗ n − ℓ | 1 � ⊗ ℓ = | 0 .. 0 → | D n 1 .. 1 � − ℓ � . ���� ���� n − ℓ ℓ “for all ℓ ≤ k ”: allows for an Inductive Approach, constructing U n , k inductively over n , allows for Symmetric State Preparation, as U n , n can be used to construct all Dicke States. Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 11
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 � : � � n � − 1 | D n ℓ � = | x � 2 ℓ x has n qubits with exactly ℓ 1’s � � n | D n − 1 n | D n − 1 ℓ n − ℓ = ℓ − 1 � ⊗ | 1 � + � ⊗ | 0 � . ℓ Assume we know how to design U n − 1 , k but do not know how to design U n , k : ℓ � should be prepared by U n , k from | 0 � ⊗ n − ℓ | 1 � ⊗ ℓ , | D n ℓ − 1 � can be prepared by U n − 1 , k from | 0 � ⊗ n − ℓ | 1 � ⊗ ℓ − 1 , | D n − 1 � can be prepared by U n − 1 , k from | 0 � ⊗ n − ℓ − 1 | 1 � ⊗ ℓ . | D n − 1 ℓ Observation : The only thing missing is a unitary which implements the mapping � � | 0 � ⊗ n − ℓ − 1 | 0 � | 1 � ⊗ ℓ − n | 0 � ⊗ n − ℓ − 1 | 0 � | 1 � ⊗ ℓ − 1 | 1 � + n | 0 � ⊗ n − ℓ − 1 | 1 � ⊗ ℓ | 0 � . n − ℓ ℓ → Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 12
Los Alamos National Laboratory Split & Cyclic Shift Unitaries Definition (Split & Cyclic Shift Unitaries SCS n , k ) A Split & Cyclic Shift Unitary SCS n , k is any unitary which for all ℓ ∈ 1 , . . . , k and k < n implements the mappings → | 0 � ⊗ k +1 , SCS n , k : | 0 � ⊗ k +1 � � SCS n , k : | 0 � ⊗ k +1 − ℓ | 1 � ⊗ ℓ → n | 0 � ⊗ k +1 − ℓ | 1 � ⊗ ℓ + n | 0 � ⊗ k − ℓ | 1 � ⊗ ℓ | 0 � , ℓ n − ℓ → | 1 � ⊗ k +1 . SCS n , k : | 1 � ⊗ k +1 SCS n , k unitaries can be built explicitly , we now construct U n , k unitaries inductively . Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 13
Los Alamos National Laboratory Review: Gate Overview Goal : Build U n , k from SCS n , k (acting on the last k + 1 qubits) and U n − 1 , k (acting on the first n − 1 qubits): U n − 1 , k : | 0 .. 0 � �→ | D n − 1 SCS n , k : | 0 .. 0 � �→ | 00 .. 000 � � 0 � � | 0 .. 000 .. 01 � �→ | D n − 1 1 n − 1 | 00 .. 001 � �→ n | 00 .. 001 � + n | 00 .. 010 � � 1 � � 2 n − 2 | 0 .. 000 .. 11 � �→ | D n − 1 | 00 .. 011 � �→ n | 00 .. 011 � + n | 00 .. 110 � � 2 . . . . . � � . | 0 .. 001 .. 11 � �→ | D n − 1 k n − k | 01 .. 111 � �→ n | 01 .. 111 � + | 11 .. 110 � k − 1 � n � �→ | D n − 1 | 11 .. 111 � �→ | 11 .. 111 � | 0 .. 0 11 .. 11 � � �� � ���� � �� � k k +1 n − 1 − k k Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 14
Los Alamos National Laboratory Inductive Construction of U n , k The Split & Cyclic Shift unitaries SCS n , k act non-trivially only on the last k + 1 of n qubits, and preceding U n − 1 , k by SCS n , k we get U n , k . | 0 � ⊗ n − k − 1 SCS 2 , 1 SCS 3 , 2 . . . SCS k +1 ,k = U n − 1 ,k = n − k : | � . . . U n,k SCS n - 1 ,k Split & . . k + 1 . Cyclic Shift SCS n,k n − 1: | � SCS n,k n : | � Inductively applying this idea, we construct U n , k by concatenating SCS n , k , SCS n − 1 , k , . . . , SCS k +1 , k as a prefix to U k , k , construct U k , k by concatenating SCS k , k − 1 , SCS k − 1 , k − 2 , . . . , SCS 2 , 1 . Andreas B¨ artschi, NSEC/CNLS, baertschi@lanl.gov August 2019 | 15
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