Hamiltonian complexity meets derandomization Alex Bredariol Grilo joint work with Dorit Aharonov
Randomness helps... Communication complexity Query complexity Cryptography Hamiltonian complexity meets derandomization 2 / 22
... in all cases? Under believable assumptions, randomness does not increase computational power It should be true, but how to prove it? Hamiltonian complexity meets derandomization 3 / 22
A glimpse of its hardness Polynomial identity testing problem Input: A representation of a polynomial p : F n → F of degree d ( n ) Output: Yes iff ∀ x 1 , ..., x n ∈ F , p ( x 1 , ..., x n ) = 0 Simple randomized algorithm ◮ Pick x 1 , ..., x n uniformly at random from a finite set S ⊆ F ◮ If p � = 0, Pr [ p ( x 1 , ..., x n ) = 0] ≤ d | S | How to find such “witness” deterministically? Hamiltonian complexity meets derandomization 4 / 22
MA vs. NP Hamiltonian complexity meets derandomization 5 / 22
MA vs. NP Problem L ∈ NP x 0 / 1 D y for x ∈ L yes , ∃ y D ( x , y ) = 1 for x ∈ L no , ∀ y D ( x , y ) = 0 Hamiltonian complexity meets derandomization 5 / 22
MA vs. NP Problem L ∈ NP Problem L ∈ MA x x 0 / 1 0 / 1 D R y y for x ∈ L yes , for x ∈ L yes , ∃ y Pr [ R ( x , y ) = 1] ≥ 2 ∃ y D ( x , y ) = 1 3 for x ∈ L no , for x ∈ L no , ∀ y Pr [ R ( x , y ) = 0] ≥ 2 ∀ y D ( x , y ) = 0 3 Hamiltonian complexity meets derandomization 5 / 22
MA vs. NP Problem L ∈ NP Problem L ∈ MA x x 0 / 1 0 / 1 D R y y for x ∈ L yes , for x ∈ L yes , ∃ y D ( x , y ) = 1 ∃ y Pr [ R ( x , y ) = 1] = 1 for x ∈ L no , for x ∈ L no , ∀ y Pr [ R ( x , y ) = 0] ≥ 2 ∀ y D ( x , y ) = 0 3 Hamiltonian complexity meets derandomization 5 / 22
MA vs. NP Problem L ∈ NP Problem L ∈ MA x x 0 / 1 0 / 1 D R y y for x ∈ L yes , for x ∈ L yes , ∃ y D ( x , y ) = 1 ∃ y Pr [ R ( x , y ) = 1] = 1 for x ∈ L no , for x ∈ L no , ∀ y Pr [ R ( x , y ) = 0] ≥ 2 ∀ y D ( x , y ) = 0 3 Derandomization conjecture MA = NP Hamiltonian complexity meets derandomization 5 / 22
Hamiltonian complexity Physical systems are described by Hamiltonians Hamiltonian complexity meets derandomization 6 / 22
Hamiltonian complexity Physical systems are described by Hamiltonians Find configurations that minimize energy of a system Groundstates of Hamiltonians Hamiltonian complexity meets derandomization 6 / 22
Hamiltonian complexity Physical systems are described by Hamiltonians Find configurations that minimize energy of a system Groundstates of Hamiltonians Interactions are local Hamiltonian complexity meets derandomization 6 / 22
Hamiltonian complexity Physical systems are described by Hamiltonians Find configurations that minimize energy of a system Groundstates of Hamiltonians Interactions are local Look this problem through lens of TCS Hamiltonian complexity meets derandomization 6 / 22
Hamiltonian complexity Physical systems are described by Hamiltonians Find configurations that minimize energy of a system Groundstates of Hamiltonians Interactions are local Look this problem through lens of TCS Local Hamiltonian problem ( k -LH α,β ) Input: Local Hamiltonians H 1 , ... H m , each acting on k out of a n -qubit system; H = � i H i yes-instance: � ψ | H | ψ � ≤ α m for some | ψ � no-instance: � ψ | H | ψ � ≥ β m for all | ψ � Hamiltonian complexity meets derandomization 6 / 22
Hamiltonian complexity Physical systems are described by Hamiltonians Find configurations that minimize energy of a system Groundstates of Hamiltonians Interactions are local Look this problem through lens of TCS Local Hamiltonian problem ( k -LH α,β ) Input: Local Hamiltonians H 1 , ... H m , each acting on k out of a n -qubit system; H = � i H i yes-instance: � ψ | H | ψ � ≤ α m for some | ψ � no-instance: � ψ | H | ψ � ≥ β m for all | ψ � How hard is this problem? Hamiltonian complexity meets derandomization 6 / 22
Restrictions on the Hamiltonians Local Hamiltonian H = � i H i is called stoquastic if the off-diagonal elements of each H i are non-positive Hamiltonian complexity meets derandomization 7 / 22
Restrictions on the Hamiltonians Local Hamiltonian H = � i H i is called stoquastic if the off-diagonal elements of each H i are non-positive This definition is basis dependent. Hamiltonian complexity meets derandomization 7 / 22
Restrictions on the Hamiltonians Local Hamiltonian H = � i H i is called stoquastic if the off-diagonal elements of each H i are non-positive This definition is basis dependent. Projector P i onto the groundspace of H i Hamiltonian complexity meets derandomization 7 / 22
Restrictions on the Hamiltonians Local Hamiltonian H = � i H i is called stoquastic if the off-diagonal elements of each H i are non-positive This definition is basis dependent. Projector P i onto the groundspace of H i ◮ P i = � j | φ i , j �� φ i , j | Hamiltonian complexity meets derandomization 7 / 22
Restrictions on the Hamiltonians Local Hamiltonian H = � i H i is called stoquastic if the off-diagonal elements of each H i are non-positive This definition is basis dependent. Projector P i onto the groundspace of H i ◮ P i = � j | φ i , j �� φ i , j | ◮ � φ i , j | φ i , j ′ � = 0, for j � = j ′ Hamiltonian complexity meets derandomization 7 / 22
Restrictions on the Hamiltonians Local Hamiltonian H = � i H i is called stoquastic if the off-diagonal elements of each H i are non-positive This definition is basis dependent. Projector P i onto the groundspace of H i ◮ P i = � j | φ i , j �� φ i , j | ◮ � φ i , j | φ i , j ′ � = 0, for j � = j ′ ◮ | φ i , j � have real non-negative amplitudes. Hamiltonian complexity meets derandomization 7 / 22
Restrictions on the Hamiltonians Local Hamiltonian H = � i H i is called stoquastic if the off-diagonal elements of each H i are non-positive This definition is basis dependent. Projector P i onto the groundspace of H i ◮ P i = � j | φ i , j �� φ i , j | ◮ � φ i , j | φ i , j ′ � = 0, for j � = j ′ ◮ | φ i , j � have real non-negative amplitudes. x α x | x � , α x ∈ R + Groundstate | ψ � = � Hamiltonian complexity meets derandomization 7 / 22
Restrictions on the Hamiltonians Local Hamiltonian H = � i H i is called stoquastic if the off-diagonal elements of each H i are non-positive This definition is basis dependent. Projector P i onto the groundspace of H i ◮ P i = � j | φ i , j �� φ i , j | ◮ � φ i , j | φ i , j ′ � = 0, for j � = j ′ ◮ | φ i , j � have real non-negative amplitudes. x α x | x � , α x ∈ R + Groundstate | ψ � = � In this work: | φ i , j � = | T i , j � , where T i , j ⊆ { 0 , 1 } k Hamiltonian complexity meets derandomization 7 / 22
Stoquastic Hamiltonian problem Uniform stoquastic local Hamiltonian problem Input: Uniform stoquastic local Hamiltonians H 1 , ... H m , each acting on k out of a n -qubit system; H = � i H i yes-instance: � ψ | H | ψ � = 0 no-instance: � ψ | H | ψ � ≥ β m for all | ψ � Hamiltonian complexity meets derandomization 8 / 22
Stoquastic Hamiltonian problem Uniform stoquastic local Hamiltonian problem Input: Uniform stoquastic local Hamiltonians H 1 , ... H m , each acting on k out of a n -qubit system; H = � i H i yes-instance: � ψ | H | ψ � = 0 no-instance: � ψ | H | ψ � ≥ β m for all | ψ � 1 for some β = poly ( n ) , it is MA-complete (Bravyi-Terhal ’08) Hamiltonian complexity meets derandomization 8 / 22
Stoquastic Hamiltonian problem Uniform stoquastic local Hamiltonian problem Input: Uniform stoquastic local Hamiltonians H 1 , ... H m , each acting on k out of a n -qubit system; H = � i H i yes-instance: � ψ | H | ψ � = 0 no-instance: � ψ | H | ψ � ≥ β m for all | ψ � 1 for some β = poly ( n ) , it is MA-complete (Bravyi-Terhal ’08) Our work: if β is constant, it is in NP Hamiltonian complexity meets derandomization 8 / 22
Outline Connection between Hamiltonian complexity and derandomization 1 MA and stoquastic Hamiltonians 2 Proof sketch 3 Open problems 4 Hamiltonian complexity meets derandomization 9 / 22
Back to NP vs. MA Theorem (BT ’08) Deciding if Unif. Stoq. LH is frustration-free or inverse polynomial frustrated is MA-complete. Theorem (This work) Deciding if Unif. Stoq. LH is frustration-free or constant frustrated is NP-complete. Hamiltonian complexity meets derandomization 10 / 22
Back to NP vs. MA Corollary Suppose a deterministic polynomial-time map φ ( H ) = H ′ such that Hamiltonian complexity meets derandomization 11 / 22
Back to NP vs. MA Corollary Suppose a deterministic polynomial-time map φ ( H ) = H ′ such that 1 H ′ is a uniform stoquastic Hamiltonian with constant locality and degree; Hamiltonian complexity meets derandomization 11 / 22
Back to NP vs. MA Corollary Suppose a deterministic polynomial-time map φ ( H ) = H ′ such that 1 H ′ is a uniform stoquastic Hamiltonian with constant locality and degree; 2 if H is frustration-free, H ′ is frustration free; Hamiltonian complexity meets derandomization 11 / 22
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