Quantum supremacy with random circuits Christian Majenz QMATH, - - PowerPoint PPT Presentation
Quantum supremacy with random circuits Christian Majenz QMATH, University of Copenhagen Joint work (in progress) with Gorjan Alagic, QMATH, and Bill Fefferman, University of Maryland USC physics seminar, University of Southern California, Los
Christian Majenz
QMATH, University of Copenhagen Joint work (in progress) with Gorjan Alagic, QMATH, and Bill Fefferman, University of Maryland USC physics seminar, University of Southern California, Los Angeles
03/10/2017
1 / 24
1 / 24
2 / 24
2 / 24
2 / 24
2 / 24
2 / 24
2 / 24
2 / 24
Caveats:
3 / 24
Caveats:
◮ Only Grover search is well-founded speed-up, and only in the
query complexity model
3 / 24
Caveats:
◮ Only Grover search is well-founded speed-up, and only in the
query complexity model
◮ We don’t have a quantum computer!
3 / 24
4 / 24
”we” being theorists:
4 / 24
”we” being theorists: (*) Time complexity separations under standard assumptions like P = NP & friends
4 / 24
”we” being theorists: (*) Time complexity separations under standard assumptions like P = NP & friends
◮ ”Quantum supremacy”
4 / 24
”we” being theorists: (*) Time complexity separations under standard assumptions like P = NP & friends
◮ ”Quantum supremacy”
4 / 24
”we” being theorists: (*) Time complexity separations under standard assumptions like P = NP & friends
◮ ”Quantum supremacy”
”we” being experimenters:
4 / 24
”we” being theorists: (*) Time complexity separations under standard assumptions like P = NP & friends
◮ ”Quantum supremacy”
”we” being experimenters:
◮ Computational tasks for proof-of-principle grade machines
that will convince quantum computing doubters and funding agencies
4 / 24
”we” being theorists: (*) Time complexity separations under standard assumptions like P = NP & friends
◮ ”Quantum supremacy”
”we” being experimenters:
◮ Computational tasks for proof-of-principle grade machines
that will convince quantum computing doubters and funding agencies ⇒ Find (*) with minimal experimental requirements, best w/o requiring fault-tolerant quantum computation.
4 / 24
Motivation: Quantum computers and what to do with them Introduction Complexity Review: supremacy results Ingredients for supremacy Stockmeyer’s algorithm ♯P-hardness, but what kind? Supremacy with circuit families How to prove supremacy? New results
5 / 24
6 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
7 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
◮ n variables x = (x1, ..., xn) ∈ {0, 1}n, ”binary string”
7 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
◮ n variables x = (x1, ..., xn) ∈ {0, 1}n, ”binary string” ◮ Boolean formula: e.g. f (x) = m i=1(xi1 ∨ xi2 ∨ xi3)
7 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
◮ n variables x = (x1, ..., xn) ∈ {0, 1}n, ”binary string” ◮ Boolean formula: e.g. f (x) = m i=1(xi1 ∨ xi2 ∨ xi3) ◮ Calculating f (x) for given x is in P, polynomial time
7 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
◮ n variables x = (x1, ..., xn) ∈ {0, 1}n, ”binary string” ◮ Boolean formula: e.g. f (x) = m i=1(xi1 ∨ xi2 ∨ xi3) ◮ Calculating f (x) for given x is in P, polynomial time ◮ NP = Σ1: Problems as hard as deciding whether
∃x ∈ {0, 1}n : f (x)
7 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
◮ n variables x = (x1, ..., xn) ∈ {0, 1}n, ”binary string” ◮ Boolean formula: e.g. f (x) = m i=1(xi1 ∨ xi2 ∨ xi3) ◮ Calculating f (x) for given x is in P, polynomial time ◮ NP = Σ1: Problems as hard as deciding whether
∃x ∈ {0, 1}n : f (x)
◮ coNP = Π1: Problems as hard as deciding whether
f (x)∀x ∈ {0, 1}n
7 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
◮ n variables x = (x1, ..., xn) ∈ {0, 1}n, ”binary string” ◮ Boolean formula: e.g. f (x) = m i=1(xi1 ∨ xi2 ∨ xi3) ◮ Calculating f (x) for given x is in P, polynomial time ◮ NP = Σ1: Problems as hard as deciding whether
∃x ∈ {0, 1}n : f (x)
◮ coNP = Π1: Problems as hard as deciding whether
f (x)∀x ∈ {0, 1}n
◮ for x and y binary strings, write xy = (xi, ..., xn, y1, ..., yl)
7 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
◮ n variables x = (x1, ..., xn) ∈ {0, 1}n, ”binary string” ◮ Boolean formula: e.g. f (x) = m i=1(xi1 ∨ xi2 ∨ xi3) ◮ Calculating f (x) for given x is in P, polynomial time ◮ NP = Σ1: Problems as hard as deciding whether
∃x ∈ {0, 1}n : f (x)
◮ coNP = Π1: Problems as hard as deciding whether
f (x)∀x ∈ {0, 1}n
◮ for x and y binary strings, write xy = (xi, ..., xn, y1, ..., yl) ◮ Σ2: Problems as hard as deciding whether
∃x ∈ {0, 1}k : ∀y ∈ {0, 1}n−kf (xy)
7 / 24
◮ Boolean formulas provide characterizing problems for many
complexity classes:
◮ n variables x = (x1, ..., xn) ∈ {0, 1}n, ”binary string” ◮ Boolean formula: e.g. f (x) = m i=1(xi1 ∨ xi2 ∨ xi3) ◮ Calculating f (x) for given x is in P, polynomial time ◮ NP = Σ1: Problems as hard as deciding whether
∃x ∈ {0, 1}n : f (x)
◮ coNP = Π1: Problems as hard as deciding whether
f (x)∀x ∈ {0, 1}n
◮ for x and y binary strings, write xy = (xi, ..., xn, y1, ..., yl) ◮ Σ2: Problems as hard as deciding whether
∃x ∈ {0, 1}k : ∀y ∈ {0, 1}n−kf (xy) ...
7 / 24
8 / 24
8 / 24
8 / 24
8 / 24
8 / 24
8 / 24
8 / 24
8 / 24
8 / 24
8 / 24
◮ ♯P: Problems as hard as finding |{x ∈ {0, 1}n|f (x)}|.
9 / 24
◮ ♯P: Problems as hard as finding |{x ∈ {0, 1}n|f (x)}|. ◮ PH ⊂ ♯P (Toda, ’89)
9 / 24
◮ ♯P: Problems as hard as finding |{x ∈ {0, 1}n|f (x)}|. ◮ PH ⊂ ♯P (Toda, ’89) ◮ C hard problem L0: for all problems L ∈ C, L can be solved by
solving L0.
9 / 24
◮ ♯P: Problems as hard as finding |{x ∈ {0, 1}n|f (x)}|. ◮ PH ⊂ ♯P (Toda, ’89) ◮ C hard problem L0: for all problems L ∈ C, L can be solved by
solving L0. → polynomial time reduction
9 / 24
◮ ♯P: Problems as hard as finding |{x ∈ {0, 1}n|f (x)}|. ◮ PH ⊂ ♯P (Toda, ’89) ◮ C hard problem L0: for all problems L ∈ C, L can be solved by
solving L0. → polynomial time reduction
◮ Examples:
9 / 24
◮ ♯P: Problems as hard as finding |{x ∈ {0, 1}n|f (x)}|. ◮ PH ⊂ ♯P (Toda, ’89) ◮ C hard problem L0: for all problems L ∈ C, L can be solved by
solving L0. → polynomial time reduction
◮ Examples:
· NP hard: Travelling salesman, subset sum, satisfiability
9 / 24
◮ ♯P: Problems as hard as finding |{x ∈ {0, 1}n|f (x)}|. ◮ PH ⊂ ♯P (Toda, ’89) ◮ C hard problem L0: for all problems L ∈ C, L can be solved by
solving L0. → polynomial time reduction
◮ Examples:
· NP hard: Travelling salesman, subset sum, satisfiability · ♯P hard: Calculate the permanent per of a {0, 1}-valued nxn matrix, per(A) =
n
Aiσ(i)
9 / 24
◮ There are problems where randomness seems to help
10 / 24
◮ There are problems where randomness seems to help ◮ BPP: problems solvable in polynomial time with randomness
10 / 24
◮ 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
◮ 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
◮ 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
◮ 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
◮ 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 (e-) sampling: sample from p : {0, 1}n → [0, 1].
11 / 24
◮ 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 (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 − q1 ≤ ε.
11 / 24
◮ 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 − q1 ≤ ε. ! Only additively approximate sampling problems can be solved
11 / 24
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
◮ If m-sampling for constant depth quantum circuits is easy,
then BQP ⊂ AM (Terhal and DiVincenzo ’04).
13 / 24
◮ 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
◮ 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
◮ 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
◮ 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
13 / 24
14 / 24
Recall: caculating |{x ∈ {0, 1}n|f (x)}| for Boolean formulas is ♯P hard.
15 / 24
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 Σ2
15 / 24
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 Σ2
15 / 24
16 / 24
◮ the permanent
16 / 24
◮ the permanent ◮ the partition function of certain ising models
16 / 24
◮ the permanent ◮ the partition function of certain ising models
exact approximate worst case average case per ising per per ising ?
◮ α-average case hard: can remove any fraction α of the
problem instances ⇒ still hard
16 / 24
17 / 24
◮ Set of gates Γ ⊂ U(C2 ⊗ C2)
18 / 24
◮ Set of gates Γ ⊂ U(C2 ⊗ C2) ◮ For n qbits and U ∈ Γ, Uij acts on qbits i and j, Γn = {Uij}
18 / 24
◮ Set of gates Γ ⊂ U(C2 ⊗ C2) ◮ For n qbits and U ∈ Γ, Uij acts on qbits i and j, Γn = {Uij} ◮ Circuit: C = (V1, ..., Vm), Vi ∈ Γn
18 / 24
◮ Set of gates Γ ⊂ U(C2 ⊗ C2) ◮ For n qbits and U ∈ Γ, Uij acts on qbits i and j, Γn = {Uij} ◮ Circuit: C = (V1, ..., Vm), Vi ∈ Γn ◮ Circuit family: C = (Cn)n∈N, Cn set of n qbit circuits
18 / 24
◮ C circuit family
19 / 24
◮ C circuit family ◮ assume calculating | 0| C |0 |2 is ♯P hard to m-approximate
19 / 24
◮ C circuit family ◮ assume calculating | 0| C |0 |2 is ♯P hard to m-approximate ◮ suppose A classical m-approximate sampling algorithm for
pC(x) = | 0| C |x |2
19 / 24
◮ C circuit family ◮ assume calculating | 0| C |0 |2 is ♯P hard to m-approximate ◮ suppose A classical m-approximate sampling algorithm for
pC(x) = | 0| C |x |2
◮ A(C, r)
19 / 24
◮ C circuit family ◮ assume calculating | 0| C |0 |2 is ♯P hard to m-approximate ◮ suppose A classical m-approximate sampling algorithm for
pC(x) = | 0| C |x |2
◮ A(C, r) ◮ define f (r) = 1 if A(C, r) = 0, f (r) = 0 else
19 / 24
◮ C circuit family ◮ assume calculating | 0| C |0 |2 is ♯P hard to m-approximate ◮ suppose A classical m-approximate sampling algorithm for
pC(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 Σ2 ⇒ collapse!
19 / 24
! we wanted supremacy for additive approximation
20 / 24
! we wanted supremacy for additive approximation
◮ C circuit family s.t. C ∈ C ⇒ C(σx)i ∈ C for all i.
20 / 24
! we wanted supremacy for additive approximation
◮ C circuit family s.t. C ∈ C ⇒ C(σx)i ∈ C for all i. ◮ assume calculating | 0| C |0 |2 is average case ♯P hard to
m-approximate
20 / 24
! we wanted supremacy for additive approximation
◮ C circuit family s.t. C ∈ C ⇒ C(σx)i ∈ C for all i. ◮ assume calculating | 0| C |0 |2 is average case ♯P hard to
m-approximate
◮ Assume EC∈RCn[| 0| C |0 |4] ≤ β2−2n ⇒ most | 0| C |0 | are
not too small
20 / 24
! we wanted supremacy for additive approximation
◮ C circuit family s.t. C ∈ C ⇒ C(σx)i ∈ C for all i. ◮ assume calculating | 0| C |0 |2 is average case ♯P hard to
m-approximate
◮ Assume EC∈RCn[| 0| C |0 |4] ≤ β2−2n ⇒ most | 0| C |0 | are
not too small
◮ suppose A classical a-approximate sampling algorithm
20 / 24
! we wanted supremacy for additive approximation
◮ C circuit family s.t. C ∈ C ⇒ C(σx)i ∈ C for all i. ◮ assume calculating | 0| C |0 |2 is average case ♯P hard to
m-approximate
◮ Assume EC∈RCn[| 0| C |0 |4] ≤ β2−2n ⇒ 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
! we wanted supremacy for additive approximation
◮ C circuit family s.t. C ∈ C ⇒ C(σx)i ∈ C for all i. ◮ assume calculating | 0| C |0 |2 is average case ♯P hard to
m-approximate
◮ Assume EC∈RCn[| 0| C |0 |4] ≤ β2−2n ⇒ 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
◮ low relative error for most C
21 / 24
◮ possibly high relative error for most C
21 / 24
◮ low relative error for most C
21 / 24
◮ Bremner et al.’s multiplicative to additive reduction can be
formulated as a flexible lemma
22 / 24
◮ 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
◮ 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
◮ 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
◮ 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
we need hardness of approximate sampling!
realistic supremacy results need strong unproven hardness assumptions approximate sampling for random cliffords on product states is likely hard
23 / 24
24 / 24