Now we show that it is also #P-hard to computePermanent2 on most matrices over a suffi- ciently large finite field. An analogous argument works for thePermanentfunction.
Theorem 25(Lipton [Lip91]). If there is a randomized polynomial time algorithmOsuch that:
Pr
X
O(X) =Permanent2[X]
>1−1/(6n+ 3)
With each element inXchosen uniformly at random from a finite fieldFof size at least2n+1, then we can useO at most apoly(n)number of times to computePermanent2[X]for anyX ∈Fn×n in randomizedpoly(n)time.
Proof. The proof uses only that thePermanent2 function is of degree2n. We need to compute the Permanent2[X] for an arbitraryX ∈ Fn×n. Consider a procedure that chooses a random Y ∈ Fn×n and then picks an arbitrary subsetS ⊆ F of cardinality2n+ 1 which doesn’t include 0. For each s ∈ S, compute the valueas = O(X +sY). Use Lagrange linear interpolation to compute the unique polynomial p(s)of degree2n such that p(s) = as for alls ∈ S, and output p(0). First we note that sinceY is chosen uniformly at random fromFn×n, it is clear that, for each s,X+sY is uniformly distributed overFn×n. As a direct consequence of this we know:
Lemma 26. For eachX,
Pr
Y
∀s ∈S :p(s) =Permanent2[X+sY]
>2/3
Proof. For eachX, invoking a union bound, we have:
Pr
Y
∃s ∈S :Permanent2[X+sY]6=p(s)
≤X
s∈S
Pr
Permanent2[X+sY]6=p(s)
≤ 2n+ 1
6n+ 3 = 1/3
Where the last inequality comes from the error probability ofO.
Now conditioned onp(s) = Permanent2[X+sY]for alls∈Sit follows thatp(0) =Permanent2[X]
because, for fixedXandY the univariate polynomialf(s) =Permanent2[X+sY]andp(s)are of degree2nand agree on2n+ 1points.
Now we will show that we can prove this hardness result even when the matrix values are Real and the entries are distributed from an “autocorrelatable” distribution.
Definition 27(Autocorrelatable distribution). We say a continuous distributionDoverRis auto- correlatable if there exists some constantc >0so that for all >0andz ∈[−1,1]:
∞
Z
−∞
|D(x)− D((1−)x+z)| dx≤c
For example, Aaronson and Arkhipov [AA13] have shown that the Gaussian Distribution with mean 0 and variance 1 is autocorrelatable.
Theorem 28(Generalizing [AA13]). SupposeDis some autocorrelatable distribution overRthat can be sampled efficiently, and we are given an oracleOso that:
Pr
Y∼Dn×n[O(Y) =Permanent2[Y]]≥3/4 +δ
for someδ = 1/poly(n). Then given anX ∈ [−1,1]n×n, we can useO at mostpoly(n)times to computePermanent2[X].
Proof. First we cite the Berlekamp-Welch algorithm on noisy interpolation of univariate polyno- mials over arbitrary fields,F:
Theorem 29(Berlekamp-Welch [Ber84]). Letq be a univariate polynomial of degreedover any fieldF. Suppose we are givenmdistinct pairs of F-elements(x1, y1),(x2, y2)...(xm, ym)and are
promised thatq(xi) = yi for at least m+d2 values ofi. There exists a deterministic algorithm to reconstructqusingpoly(n, d)field operations.
Given X ∈ [−1,+1]n×n, we choose Y ∼ Dn×n and let X(t) = (1− t)Y +tX. Note that X(1) =X andX(0) = Y. We define the univariate polynomialq(t)of degree at most2n, so that q(t) =Permanent[X(t)]2.
We can no longer guarantee that the matrixX(t)is distributed from Dn×n, but for small enough values oft, we can show that the distribution it is drawn from is close in statistical distance.
Now letS = 2nδ and= 2cSnδ 2. For everys∈[S]we defineas =O(X(s)). Our goal is to use the Berlekamp-Welch algorithm, Theorem 29, to recoverq(t), using the noisy evaluations as. Then we simply returnq(1) =Permanent2[X]as desired.
We know by definition:
Pr
Y∼Dn×n[O(Y) =Permanent[Y]2]≥3/4 +δ Then if we letDs be the distribution the matrixX(s)is drawn from, we have:
Pr[O(X(s)) =q(s)]≥3/4 +δ− kDn×n− Dsk ≥3/4 +δ−cSn2 = 3/4 +δ/2
Where the second inequality follows from the triangle inequality and the fact thatD is autocorre- latable.
LetT be the set ofsso thatq(s) = as. By Markov:
Pr
|T| ≥ 1
2 +δ 2
S
≥1−
1 4 −2δ
1
2 −2δ ≥ 1 2 +δ
2 Since S = 2n/δ we have that 12 +δ2
S = S+2n2 = m+d2 when the number of sampled points m = S and the degreed = 2n and so we can use Berlekamp-Welch algorithm from Theorem 29
to computePermanent2[X]whenever|T| ≥ 12 + δ2 S.
By repeatingO(1/δ2)times for different choices ofY and taking the majority vote, we can com- putePermanent2[X].
We now summarize these results in a table, which describes known hardness results for thePermanent2 (or equivalently, thePermanent) function.
Approximation Entries in Matrix Success Probability #P-hard?
exact {0,1}orZ 1 Yes! Thm. 13
-multiplicative Z+ 1 Easy [JSV04]
-multiplicative Z 1 Yes! Thm. 15
exact Fp,p= 2n+ 1 1−1/(6n+ 3) Yes! Thm. 25
exact R 3/4 + 1/poly(n)(over autocorr. dist.) Yes! Thm. 28
-multiplicative R 1−1/poly(n) ?
We also note that the Lipton result, Theorem 25, is known to be true for much stronger set- tings of parameters, using more sophisticated interpolation techniques (see [AA13] for a complete overview.)
Chapter 4
The Power of Exact Quantum Sampling
In this section we prove that, unless thePH collapses to a finite level, there is a class of distri- butions that can be sampled efficiently on a Quantum Computer, that cannot be sampled exactly classically. To do this we (again) cite a theorem by Stockmeyer on the ability to “approximate count” inside thePH.
Theorem 30 (Stockmeyer [Sto85]). Given as input a function f : {0,1}n → {0,1}m and y ∈ {0,1}m, there is a procedure that outputsαsuch that:
(1−) Pr
x∼U{0,1}n
[f(x) = y]≤α≤(1 +) Pr
x∼U{0,1}n
[f(x) =y]
In randomized timepoly(n,1/)with access to anNPoracle.
Note that as a consequence of Theorem 30, given an efficiently computablef : {0,1}n → {0,1}
we can compute a multiplicative approximation toPrx∼U{0,1}n[f(x) = 1] =
P
x∈{0,1}nf(x)
2n in thePH.
Despite this, we have shown, as a consequence of Theorem 22, that the same multiplicative ap- proximation becomes#P-hard iff is{±1}-valued.
Now we show the promised class of quantumly sampleable distributions:
Definition 31 (Df). Givenf : {0,1}n → {±1}, we define the distribution Df over {0,1}n as
follows:
DPrf,n[y] =
P
x∈{0,1}n
(−1)hx,yif(x)
!2
22n
The fact that this is a distribution will follow from the proceeding discussion.
Theorem 32. For all efficiently computable f : {0,1}n → {±1} we can sample from Df in poly(n)time on a Quantum Computer.
Proof. Consider the following quantum algorithm:
1. Prepare the state 2n/21
P
x∈{0,1}n
|xi
2. Since by assumptionf is efficiently computable, we can applyf to the phases (as discussed in Section 2.3), with two quantum queries tof resulting in:
|fi= 1 2n/2
X
x∈{0,1}n
f(x)|xi
3. Apply thenqubit Hadamard,H⊗n 4. Measure in the standard basis
Note thatH⊗n|fi = 21n
P
y∈{0,1}n
P
x∈{0,1}n
(−1)hx,yif(x)|yiand therefore the distribution sampled by the above quantum algorithm isDf.
As before, the key observation is that(h00...0|H⊗n|fi)2 =
P
x∈{0,1}nf(x)
!2
22n , and therefore encodes a#P-hard quantity in an exponentially small amplitude. We can exploit this hardness classically if we assume the existence of a classical sampler, which we define to mean an efficient random algorithm whose output is distributed via this distribution.
Theorem 33 (Folklore, e.g., [Aar11]). Suppose we have a classical randomized algorithm B, which given as input0n, samples fromDf in timepoly(n), then thePHcollapses toBPPNP.
Proof. The proof follows by applying Theorem 30 to obtain an approximate count to the fraction of random stringsrso thatB(0n, r) = 00..0. Formally, we can output anαso that:
(1−)
P
x∈{0,1}n
f(x)
!2
22n ≤α≤
P
x∈{0,1}n
f(x)
!2
22n (1 +)
In timepoly(n,1/)using anNPoracle. Multiplying through by22nallows us to get a multiplica- tive approximation to P
x∈{0,1}n
f(x)
!2
in thePH. This task is#P-hard, as proven in Theorem 23. Since we know by Theorem 12,PH ⊆ P#P, we now have thatP#P ⊆ BPPNP ⇒PH ⊆ BPPNP leading to our theorem. Note that this theorem would hold even under the weaker as- sumption that the sampler is contained inBPPPH.
We end this Chapter by noting that Theorem 33 is extremely sensitive to the exactness condition imposed on the classical sampler, because the amplitude of the quantum state on which we based our hardness is only exponentially small. Thus it is clear that by weakening our sampler to an
“approximate” setting in which the sampler is free to sample any distribution Y so that the Total Variation distancekY−Dfk ≤1/poly(n)we no longer can guarantee any complexity consequence using the above construction. Indeed, this observation makes the construction quite weak– for instance, it may even be unfair to demand that any physical realization of this quantum circuit itselfsamples exactly from this distribution! In the proceeding sections we are motivated by this apparent weakness and discuss the intractability of approximately sampling in this manner from quantumly sampleable distributions.