Shor's famous factoring algorithm [Sho94] provides an example of a problem that can be solved efficiently on a quantum computer without any known efficient classical algorithm. In particular, there is a problem that can be solved efficiently on a quantum computer that cannot be solved efficiently using non-determinism.
Overview
In Section 5.2 we define a general class of distributions that can be sampled exactly on a quantum computer. We then show in Section 5.3 that the existence of an approximate classical sampler for these distributions implies the existence of an additive approximate mean-case solution to the efficiently specifiable polynomial.
Computational Complexity Basics
Then define ΣPk recursively such that for k > 0, ΣPk+1 = NPΣPk, where this notation refers to the class of decision problems decidable in NP with the ability to query an oracle that decides any problem in ΣPk. Where ∃Si is the notation meaning "there exists an assignment to the variables in S1", ∀Sj is the notation meaning "for all assignments to the variables inSj", and Qkis the k-th quantifier.
Quantum Preliminaries
We define {Πrj} to be a collection of projection operators projecting into the subspace spanned by the designated|vjifor allvj associated with the same output value rj. And Toffoli is the three qubit gate that implements a Controlled-Controlled-Not, which simply flips the state of the last qubit if the first two qubits are 1.
Quantum Complexity and BQP
Note that this immediately implies BQP ⊆ PSPACE, because we can simply compute the value of each path in the sum of 2.1 using only a poly(n) amount of space. We can appeal to the bounds (and the characterization of P#P) proved in [FFK91] to show that it is sufficient to prove BQP⊆P#P.
Better Classical Algorithms for Simulating BQP using Ap- proximate Counting?
Note that the gate set {Rθ, T} is a universal gate set for quantum computing since we can always . Then, for fixed θ > 0, we can simulate a generic quantum circuit with poly(n)Toffoli gates and r(n)Rθ gates, in time poly(n,1/) =poly.
Basic Counting Definitions and Results
We also know that calculating the Permanent of an n × n matrix with values in {0,1} is #P-complete. The fact thatPermanentis in#P can be shown by the well-known equivalence between thePermanent of a{0,1}matrix and counting the number of perfect matches in a bipartite graph. Deciding whether there is perfect matching in a bipartite graph is done in NP (and in fact in P by the Hopcroft-Karp algorithm [HK73]) and so counting the number of perfect matchings is a #P problem.
Valiant showed that counting the number of perfect matches in a bipartite graph is also #Phard, and therefore as hard as the #SAT.
The Hardness of Multiplicative Estimation of the Perma- nent
Given the ability to compute this sum for a circuit of value {±1}, we will show that we can obtain P. Given as input the description of a classical efficient circuit Cf computing a function f : {0,1 }n → {±1} we can efficiently obtain a matrix X so that we can efficiently find P. Putting all the claims together, we prove Lemma 18 and conclude that if we can compute the Permanent of an arbitrary matrix, we have an efficient classical algorithm that uses this ability to compute P.
We can also use it to show our desired result, namely the computation of a multiplicative estimate to the Permanentis#P-hard. Note that if we can compute the estimate guaranteed in the statement of the theorem, we can certainly compute sogn(Permanent[X]) = Permanent[X]. First, we note that by adding extra input bits to the circuit Cf, we can create a circuit Cf(k) such that P.
Note that using the same method as in Lemma 18, we have established that given as input the description of any classical circuit Cf that computes a function f :{0,1}n→ {±1} we can.
The Hardness of Computing the Permanent over F on Most Matrices
We will now show that we can prove this hardness result even if the matrix values are real and the entries are distributed from an “autocorrelation” distribution. We can no longer guarantee that the matrix X(t) is distributed from Dn×n, but for sufficiently small values oft we can show that the distribution from which it is drawn is close to the statistical distance. S = S+2n2 = m+d2 when the number of sampled points is m = S and the rate = 2n, so we can use the Berlekamp-Welch algorithm of Theorem 29.
For all efficiently computable f : {0,1}n → {±1}, we can sample from Df in poly(n)time on a quantum computer. Since f is effectively computable by assumption, we can apply f to the phases (as discussed in Section 2.3), with two quantum queries for the result: . Measure in a standard basis. 1)hx,yif(x)|yiin so the distribution sampled by the above quantum algorithm is Df. This stiffness can be classically exploited if we assume the existence of a classical sampler, which is defined as an efficient random algorithm whose output is distributed over this distribution.
Note that this theorem would hold even under the weaker as- sumption that the sampler is contained inBPPPH.
Approximate Sampling Definitions
Efficiently Specifiable Polynomial Sampling on a Quantum Computer
Given a polynomial, With monomials and ` ≤ exp(n), Can be efficiently specified in terms of, the resulting DQ, `can be sampled in poly(n) time on a quantum computer.
Classical Hardness of Efficiently Specifiable Polynomial Sam- pling
Note that the amplitude of each y-base state in the final state is proportional to the value of Q(Zy). Now, as proved in Section 5.5, we note that the variance of the distribution over C induced by an efficiently specifiable Q with m monomials, evaluated at uniformly distributed values over Tn` is m, and so the preceding Theorem 43 promised us that we this could be achieved by a Var [Q]-additive approximation of Q2, given a Sampler. Given Conjecture 1, relative to an efficiently specifiable polynomial Q and a distributionD, a Var[Q]-additive approximateδ-average case solution with respect toD, will lead to the.
Suppose that λ is, with high probability, a Var [Q]-additive approximation to |Q(X)|2, as guaranteed in the statement of the theorem. For the results in this section to be meaningful, we simply need the anti-centering conjecture to hold for some specified efficiency polynomial that is #P-hard to compute, with respect to any distribution from which we may sample (orUn , orB(0, k )n). We note that Aaronson and Arkhipov [AA13] conjecture the same statement as conjecture 1 for the special case of the Continuous function with respect to matrices with independently distributed entries from the complex Gaussian distribution of mean 0 and variance 1.
This is quite close to our conjecture for the case of the permanent function and uniformly distributed {±1}n×n = Tn×n2 matrix.
Computation of the Variance of Efficiently Specifiable Poly- nomial
In this way, we will consider their graded equivalent polynomial Q0k :Tnk2 →Which is a sum of m0 = mkd multilinear monomials, each of graded.
Permanent is Efficiently Specifiable
Consider the 'factorial representation', a number sequence, where the ith place value is associated with (i−1)!, and the sum of the digits multiplied by the respective place value is the value of the number itself. Clearly, this process can be achieved efficiently and reversed efficiently, and note that the largest value of any value at the ith place can be isi. At each step we maintain a list that we think originally contains n numbers in ascending order from 0 to 1.
Going from left to right, start with the leftmost number in the display and output the value at that position in the list, `. At each step we keep a list` which we think contains initially n numbers in order from 0ton−1. Going from left to right, start with the leftmost changing number and work out the position.
The Hamiltonian Cycle Polynomial is Efficiently Specifi- able
The resulting n-number sequence x is the n-cycle, in which the value of eachxi indicates the node to which the third node is mapped. Go from left to right, starting with the leftmost number in the then loop and output the position of that number in the list` (where we index the list starting at the 0 position).
Efficient Quantum Sampling
First take a2k×(k+ 1) rectangular matrixD˜(1)k whose rows are indexed by assignments to the variables x1, x2, .., xk ∈ {±1}columns and are the sum of the entries in each column whose eDkmonomial is in each corresponding elementary symmetric polynomial. Note that D˜k is exactly the matrix whose (i, j)-th entry is the evaluation of the symmetric polynomial j evaluated in an assignment in the i-th symmetry class. Note that the value of the symmetric polynomial in each assignment in an equivalence class is the same.
From this we conclude that the columns of the matrix L· D˜k, in which the ith row is D˜k. Then the first two non-zero entries in R, called r0, r1, correspond to the column normalization relating the zero and the first elementary symmetric polynomial, 1/√. We now compute r1, the normalization in column D˜k corresponding to the first basis symmetric polynomial.
Thus, the value of the first symmetric polynomial is the sum of these values, which for the i-equivalence class is exactly k−2i.
A Simple Example of “Squashed” QFT, for k = 2
Using our “Squashed QFT” to Quantumly Sample from Distributions of Efficiently Specifiable Polynomial Evalu-
By applying (L·D˜k·R)⊗n in place of the Quantum Fourier Transform overZn2 in Section 5.2 we can efficiently quantum sample DQ,k.
The Hardness of Classical Sampling from the Squashed Distribution
Because we can amplify the failure probability of Stockmeyer's algorithm to be inverse exponential. As mentioned before, our goal is to find a class of distributions {Dn}n>0 that can be sampled exactly in poly(n) time on a quantum computer, with the property that a (classical) sampler does not exist regarding that class of distributions,{Dn}n>0. Using the results in Sections 5.3 and 5.4, we can quantum sample from a class of distributions {DQ,k}n>0, where = poly(n) with the property that, if there is a classical Sampler relative to this class distributions, there exists a Var [Q]-additive δ-mean case solution for the Q2 function with respect to the B(0, k)n distribution.
Squashed QFT” unitary matrix, we could use the results from Sections 7.3 and 7.4 to makek as large asexp(n). However, we do know that we can classically evaluate a related fast (time nlog2n) polynomial transform by a theorem of Driscoll, Healy, and Rockmore [DJR97]. In this case we can simply invoke Toda’s Theorem (Theorem 12) to show that such a randomized classical solution would collapsePHto some finite level.
We note that both of these conjectures currently seem out of reach, because we do not have an example of a polynomial that is on average #P-hard to approximate (in either multiplicative or additive terms) in the sense we need.
