Quantum and probabilistic computation ◮ There are problems where randomness seems to help ◮ BPP: problems solvable in polynomial time with randomness ◮ BQP: problems solvable in polynomial time with a quantum computer 10 / 24
Quantum and probabilistic computation ◮ There are problems where randomness seems to help ◮ BPP: problems solvable in polynomial time with randomness ◮ BQP: problems solvable in polynomial time with a quantum computer ! randomness is included (measurement) 10 / 24
Quantum and probabilistic computation ◮ There are problems where randomness seems to help ◮ BPP: problems solvable in polynomial time with randomness ◮ BQP: problems solvable in polynomial time with a quantum computer ! randomness is included (measurement) ◮ BPP ⊂ BPQ ” ⊂ ” ♯ P (Bernstein and Vazirani, ’97) 10 / 24
Quantum and probabilistic computation ◮ There are problems where randomness seems to help ◮ BPP: problems solvable in polynomial time with randomness ◮ BQP: problems solvable in polynomial time with a quantum computer ! randomness is included (measurement) ◮ BPP ⊂ BPQ ” ⊂ ” ♯ P (Bernstein and Vazirani, ’97) ◮ Sampling problems: produce samples from a probability distribution p : { 0 , 1 } n → [0 , 1], exactly or approximately 10 / 24
Quantum and probabilistic computation ◮ There are problems where randomness seems to help ◮ BPP: problems solvable in polynomial time with randomness ◮ BQP: problems solvable in polynomial time with a quantum computer ! randomness is included (measurement) ◮ BPP ⊂ BPQ ” ⊂ ” ♯ P (Bernstein and Vazirani, ’97) ◮ Sampling problems: produce samples from a probability distribution p : { 0 , 1 } n → [0 , 1], exactly or approximately ◮ sampBPP , sampBPQ 10 / 24
Exact and approximate sampling ◮ exact (e-) sampling: sample from p : { 0 , 1 } n → [0 , 1]. 11 / 24
Exact and approximate sampling ◮ exact (e-) sampling: sample from p : { 0 , 1 } n → [0 , 1]. ◮ multiplicatively ε -approximate (m-) sampling: sample from any q such that q ( x ) ∈ [(1 − ε ) p ( x ) , (1 + ε ) p ( x )]. 11 / 24
Exact and approximate sampling ◮ exact (e-) sampling: sample from p : { 0 , 1 } n → [0 , 1]. ◮ multiplicatively ε -approximate (m-) sampling: sample from any q such that q ( x ) ∈ [(1 − ε ) p ( x ) , (1 + ε ) p ( x )]. ◮ additively ε -approximate (a-) sampling: sample from any q such that � p − q � 1 ≤ ε . 11 / 24
Exact and approximate sampling ◮ exact (e-) sampling: sample from p : { 0 , 1 } n → [0 , 1]. ◮ multiplicatively ε -approximate (m-) sampling: sample from any q such that q ( x ) ∈ [(1 − ε ) p ( x ) , (1 + ε ) p ( x )]. ◮ additively ε -approximate (a-) sampling: sample from any q such that � p − q � 1 ≤ ε . ! Only additively approximate sampling problems can be solved on a realistic quantum computer 11 / 24
Quantum supremacy from sampling problems Theorem blueprint L ∈ sampBQP , and if L ∈ sampBPP , then the PH collapses. In words, samples from a certain probability distribution can be generated efficiently with a quantum computer, but if there is an efficient way to do so classically, then a statement holds that is strongly believed to be false. 12 / 24
History of supremacy results ◮ If m-sampling for constant depth quantum circuits is easy, then BQP ⊂ AM (Terhal and DiVincenzo ’04). 13 / 24
History of supremacy results ◮ If m-sampling for constant depth quantum circuits is easy, then BQP ⊂ AM (Terhal and DiVincenzo ’04). ◮ If m-sampling for commuting quantum circuits is easy, then the PH collapses (Bremner et al. ’10). 13 / 24
History of supremacy results ◮ If m-sampling for constant depth quantum circuits is easy, then BQP ⊂ AM (Terhal and DiVincenzo ’04). ◮ If m-sampling for commuting quantum circuits is easy, then the PH collapses (Bremner et al. ’10). ◮ If a-sampling for a beam splitter network is easy, then the PH collapses under some extra assumptions (Aaronson and Arkhipov ’11). 13 / 24
History of supremacy results ◮ If m-sampling for constant depth quantum circuits is easy, then BQP ⊂ AM (Terhal and DiVincenzo ’04). ◮ If m-sampling for commuting quantum circuits is easy, then the PH collapses (Bremner et al. ’10). ◮ If a-sampling for a beam splitter network is easy, then the PH collapses under some extra assumptions (Aaronson and Arkhipov ’11). ◮ If e-sampling for Clifford circuits and general product state inputs is easy, then the PH collapses (Josza and van den Nest ’13). 13 / 24
History of supremacy results ◮ If m-sampling for constant depth quantum circuits is easy, then BQP ⊂ AM (Terhal and DiVincenzo ’04). ◮ If m-sampling for commuting quantum circuits is easy, then the PH collapses (Bremner et al. ’10). ◮ If a-sampling for a beam splitter network is easy, then the PH collapses under some extra assumptions (Aaronson and Arkhipov ’11). ◮ If e-sampling for Clifford circuits and general product state inputs is easy, then the PH collapses (Josza and van den Nest ’13). ◮ If a-sampling for commuting quantum circuits is easy, then the PH collapses under some extra assumptions (Bremner et al. ’16). 13 / 24
Ingredients for supremacy 14 / 24
Stockmeyer’s approximate counting Recall: caculating |{ x ∈ { 0 , 1 } n | f ( x ) }| for Boolean formulas is ♯ P hard. 15 / 24
Stockmeyer’s approximate counting Recall: caculating |{ x ∈ { 0 , 1 } n | f ( x ) }| for Boolean formulas is ♯ P hard. Theorem (Stockmeyer ’85) Approximating |{ x ∈ { 0 , 1 } n | f ( x ) }| up to multiplicative error ε can be done in Σ 2 15 / 24
Stockmeyer’s approximate counting Recall: caculating |{ x ∈ { 0 , 1 } n | f ( x ) }| for Boolean formulas is ♯ P hard. Theorem (Stockmeyer ’85) Approximating |{ x ∈ { 0 , 1 } n | f ( x ) }| up to multiplicative error ε can be done in Σ 2 15 / 24
♯ P hard problems 16 / 24
♯ P hard problems ◮ the permanent 16 / 24
♯ P hard problems ◮ the permanent ◮ the partition function of certain ising models 16 / 24
♯ P hard problems ◮ the permanent ◮ the partition function of certain ising models exact approximate per per worst case ising ising per ? average case ◮ α -average case hard: can remove any fraction α of the problem instances ⇒ still hard 16 / 24
Supremacy with circuit families 17 / 24
Circuit families ◮ Set of gates Γ ⊂ U ( C 2 ⊗ C 2 ) 18 / 24
Circuit families ◮ Set of gates Γ ⊂ U ( C 2 ⊗ C 2 ) ◮ For n qbits and U ∈ Γ, U ij acts on qbits i and j , Γ n = { U ij } 18 / 24
Circuit families ◮ Set of gates Γ ⊂ U ( C 2 ⊗ C 2 ) ◮ For n qbits and U ∈ Γ, U ij acts on qbits i and j , Γ n = { U ij } ◮ Circuit: C = ( V 1 , ..., V m ), V i ∈ Γ n 18 / 24
Circuit families ◮ Set of gates Γ ⊂ U ( C 2 ⊗ C 2 ) ◮ For n qbits and U ∈ Γ, U ij acts on qbits i and j , Γ n = { U ij } ◮ Circuit: C = ( V 1 , ..., V m ), V i ∈ Γ n ◮ Circuit family: C = ( C n ) n ∈ N , C n set of n qbit circuits 18 / 24
Proof ideas for supremacy I (Aaaronson& Arkhipov) ◮ C circuit family 19 / 24
Proof ideas for supremacy I (Aaaronson& Arkhipov) ◮ C circuit family ◮ assume calculating | � 0 | C | 0 � | 2 is ♯ P hard to m-approximate 19 / 24
Proof ideas for supremacy I (Aaaronson& Arkhipov) ◮ C circuit family ◮ assume calculating | � 0 | C | 0 � | 2 is ♯ P hard to m-approximate ◮ suppose A classical m-approximate sampling algorithm for p C ( x ) = | � 0 | C | x � | 2 19 / 24
Proof ideas for supremacy I (Aaaronson& Arkhipov) ◮ C circuit family ◮ assume calculating | � 0 | C | 0 � | 2 is ♯ P hard to m-approximate ◮ suppose A classical m-approximate sampling algorithm for p C ( x ) = | � 0 | C | x � | 2 ◮ A ( C , r ) 19 / 24
Proof ideas for supremacy I (Aaaronson& Arkhipov) ◮ C circuit family ◮ assume calculating | � 0 | C | 0 � | 2 is ♯ P hard to m-approximate ◮ suppose A classical m-approximate sampling algorithm for p C ( x ) = | � 0 | C | x � | 2 ◮ A ( C , r ) ◮ define f ( r ) = 1 if A ( C , r ) = 0, f ( r ) = 0 else 19 / 24
Proof ideas for supremacy I (Aaaronson& Arkhipov) ◮ C circuit family ◮ assume calculating | � 0 | C | 0 � | 2 is ♯ P hard to m-approximate ◮ suppose A classical m-approximate sampling algorithm for p C ( x ) = | � 0 | C | x � | 2 ◮ A ( C , r ) ◮ define f ( r ) = 1 if A ( C , r ) = 0, f ( r ) = 0 else ◮ Stockmeyer ⇒ approximating |{ r | f ( r ) }| is in Σ 2 ⇒ collapse! 19 / 24
Proof ideas for supremacy II (Bremner et al.) ! we wanted supremacy for additive approximation 20 / 24
Proof ideas for supremacy II (Bremner et al.) ! we wanted supremacy for additive approximation ◮ C circuit family s.t. C ∈ C ⇒ C ( σ x ) i ∈ C for all i . 20 / 24
Proof ideas for supremacy II (Bremner et al.) ! we wanted supremacy for additive approximation ◮ C circuit family s.t. C ∈ C ⇒ C ( σ x ) i ∈ C for all i . ◮ assume calculating | � 0 | C | 0 � | 2 is average case ♯ P hard to m-approximate 20 / 24
Proof ideas for supremacy II (Bremner et al.) ! we wanted supremacy for additive approximation ◮ C circuit family s.t. C ∈ C ⇒ C ( σ x ) i ∈ C for all i . ◮ assume calculating | � 0 | C | 0 � | 2 is average case ♯ P hard to m-approximate ◮ Assume E C ∈ R C n [ | � 0 | C | 0 � | 4 ] ≤ β 2 − 2 n ⇒ most | � 0 | C | 0 � | are not too small 20 / 24
Proof ideas for supremacy II (Bremner et al.) ! we wanted supremacy for additive approximation ◮ C circuit family s.t. C ∈ C ⇒ C ( σ x ) i ∈ C for all i . ◮ assume calculating | � 0 | C | 0 � | 2 is average case ♯ P hard to m-approximate ◮ Assume E C ∈ R C n [ | � 0 | C | 0 � | 4 ] ≤ β 2 − 2 n ⇒ most | � 0 | C | 0 � | are not too small ◮ suppose A classical a-approximate sampling algorithm 20 / 24
Proof ideas for supremacy II (Bremner et al.) ! we wanted supremacy for additive approximation ◮ C circuit family s.t. C ∈ C ⇒ C ( σ x ) i ∈ C for all i . ◮ assume calculating | � 0 | C | 0 � | 2 is average case ♯ P hard to m-approximate ◮ Assume E C ∈ R C n [ | � 0 | C | 0 � | 4 ] ≤ β 2 − 2 n ⇒ most | � 0 | C | 0 � | are not too small ◮ suppose A classical a-approximate sampling algorithm ◮ samples from q close to p but most p ( x ) are large 20 / 24
Proof ideas for supremacy II (Bremner et al.) ! we wanted supremacy for additive approximation ◮ C circuit family s.t. C ∈ C ⇒ C ( σ x ) i ∈ C for all i . ◮ assume calculating | � 0 | C | 0 � | 2 is average case ♯ P hard to m-approximate ◮ Assume E C ∈ R C n [ | � 0 | C | 0 � | 4 ] ≤ β 2 − 2 n ⇒ most | � 0 | C | 0 � | are not too small ◮ suppose A classical a-approximate sampling algorithm ◮ samples from q close to p but most p ( x ) are large ⇒ multiplicative approximation for at least some p ( x ) via anticoncentration inequality (Paley-Zygmund inequality) 20 / 24
In pictures... ◮ low relative error for most C 21 / 24
In pictures... ◮ possibly high relative error for most C 21 / 24
In pictures... ◮ low relative error for most C 21 / 24
Our preliminary results ◮ Bremner et al.’s multiplicative to additive reduction can be formulated as a flexible lemma 22 / 24
Our preliminary results ◮ Bremner et al.’s multiplicative to additive reduction can be formulated as a flexible lemma ◮ interplay between sampling error and average case hardness 22 / 24
Our preliminary results ◮ Bremner et al.’s multiplicative to additive reduction can be formulated as a flexible lemma ◮ interplay between sampling error and average case hardness ! Bremner et al.’s technique fails unless half of the instances of the problem are ♯ P hard 22 / 24
Our preliminary results ◮ Bremner et al.’s multiplicative to additive reduction can be formulated as a flexible lemma ◮ interplay between sampling error and average case hardness ! Bremner et al.’s technique fails unless half of the instances of the problem are ♯ P hard ◮ applies e.g. to all 2-design families 22 / 24
Our preliminary results ◮ Bremner et al.’s multiplicative to additive reduction can be formulated as a flexible lemma ◮ interplay between sampling error and average case hardness ! Bremner et al.’s technique fails unless half of the instances of the problem are ♯ P hard ◮ applies e.g. to all 2-design families ◮ hardness of a-sampling for Clifford circuits with product state inputs under assumptions as plausible as Bremner et al.’s 22 / 24
Take home messages we need hardness of approximate sampling! realistic supremacy results need strong unproven hardness assumptions approximate sampling for random cli ff ords on product states is likely hard 23 / 24
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