Electronic Colloquium on Computational Complexity, Report No. 159 (2016) New Hardness Results for the Permanent Using Linear Optics Daniel Grier∗ Luke Schaeffer† Abstract In 2011, Aaronson gave a striking proof, based on quantum linear optics, showing that the problem of computing the permanent of a matrix is #P-hard. Aaronson’s proof led naturally to hardness of approximation results for the permanent, and it was arguably simpler than Valiant’s seminal proof of the same fact in 1979. Nevertheless, it did not prove that computing the permanent was #P-hard for any class of matrices which was not previously known. In this paper, we present a collection of new results about matrix permanents that are derived primarily via these linear optical techniques. First, we show that the problem of computing the permanent of a real orthogonal matrix is #P-hard. Much like Aaronson’s original proof, this will show that even a multiplicative approximation remains #P-hard to compute. The hardness result even translates to permanents over finite fields, where the problem of computing the permanent of an orthogonal matrix is ModpP-hard in the finite field Fp4 for all primes p = 2, 3. Interestingly, this characterization is tight: in fields of characteristic 2, the permanent6 coincides with the determinant; in fields of characteristic 3, one can efficiently compute the permanent of an orthogonal matrix by a nontrivial result of Kogan. Finally, we use more elementary arguments to prove #P-hardness for the permanent of a pos- itive semidefinite matrix, which shows that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly despite the fact that they can be efficiently sampled by a classical computer. 1 Introduction The permanent of a matrix has been a central fixture in computer science ever since Valiant showed how it could efficiently encode the number of satisfying solutions to classic NP-complete constraint satisfaction problems [31]. His theory led to the formalization of many of the counting classes in complexity theory, including #P, the complexity class containing those functions which count the number of accepting paths of a polynomial-time non-deterministic Turing machine. Indeed, the power of these counting classes was later demonstrated by Toda’s celebrated theorem, which proved that every language in the polynomial hierarchy could be computed in polynomial time with only a single call to a #P oracle [25]. Let us recall the definition of the matrix permanent. Suppose A = a is an n n matrix { i,j} × over some field. The permanent of A is n Per(A)= ai,σ(i) σ Sn i=1 X∈ Y ∗MIT. Email: [email protected]. Supported by an NSF Graduate Research Fellowship under Grant No. 1122374. †MIT. Email: [email protected]. ‡Both authors were supported by the Vannevar Bush Faculty Fellowship from the US Department of Defense. Part of this research was completed while visiting UT Austin. 1 ISSN 1433-8092 where S is the group of permutations of 1, 2,...,n . Compare this to the determinant of A: n { } n det(A)= sign(σ) ai,σ(i). σ Sn i=1 X∈ Y Since we can compute the determinant in polynomial time (in fact, in NC2; see Berkowitz [7]), the apparent difference in complexity between the determinant and permanent comes down to the cancellation of terms in the determinant. In his original proof [31], Valiant casts the permanent in a combinatorial light, in terms of a directed graph rather than as a polynomial. Imagine the matrix A encodes the adjacency matrix of a weighted graph with vertices labeled 1,...,n . Each permutation σ on the vertices has a { } cycle decomposition, which partitions the vertices into a collection of cycles known as a cycle cover. n The weight of a cycle cover is the product of the edge weights of the cycles (i.e., i=1 ai,σ(i)), and then the permanent is the sum of the weights of all cycle covers of the graph. Equipped with Q this combinatorial interpretation of the permanent, Valiant constructs a graph by linking together different kinds of gadgets in such a way that some cycle covers correspond to solutions to a CNF formula, and the rest of the cycle covers cancel out. Valiant’s groundbreaking proof, while impressive, is fairly opaque and full of complicated gad- gets. A subsequent proof by Ben-Dor and Halevi [6] simplified the construction, while still relying on the cycle cover interpretation of the permanent. In 2009, Rudolph [22] noticed an important connection between quantum circuits and matrix permanents—a version of a correspondence we will use often in this paper. Rudolph cast the cycle cover arguments of Valiant into a more physics friendly language, which culminated in a direct proof that the amplitudes of a certain class of universal quantum circuits were proportional to the permanent. If he had pointed out that one could embed #P-hard problems into the amplitudes of a quantum circuit, then this would have constituted a semi-quantum proof that the permanent is #P-hard. Finally, in 2011, Aaronson [2] (independently from Rudolph) gave a completely self-contained and quantum linear optical proof that the permanent is #P-hard. Aaronson’s proof has a few advantages. First, much of the difficulty of the arguments based on complicated cycle cover gadgets is offloaded onto central, well-known theorems in linear optics and quantum computation. Second, it leads naturally to hardness of approximation results for the matrix permanent. Nevertheless, Aaronson conceded that his proof does not give any new classes of matrices for which the permanent is hard. Results. In this paper, we refine Aaronson’s linear optical proof technique and show that it can give #P-hardness results that were previously unknown for permanents of several classes of matrices. Specifically, we say that the permanent is #P-hard for a class of matrices if all functions in #P can be efficiently computed with single-call access to an oracle which computes permanents of matrices in that class. In general, the permanent is hard for a function class A if, given an oracle [1] for the permanent, A FPO . O ⊆ Our main result is a linear optical proof that the permanent of a real orthogonal matrix is #P-hard. This will have the consequence that the permanent of matrices in any of the classical Lie groups (e.g., invertible matrices, unitary matrices, symplectic matrices) is also #P-hard. This emphasizes one advantage of the linear optical approach over the combinatorial approach: since the matrix is constructed by multiplying matrices, it is much easier to handle groups of matrices and properties which are closed under matrix multiplication. These properties can be very difficult to handle in the cycle cover model. For instance, the matrices which arise from Valiant’s construction may indeed be invertible, but this seems to be more accidental than intentional, and a proof of their invertibility appears nontrivial. 2 In fact, our approach also reveals a surprising connection between the hardness of the permanent of orthogonal matrices over finite fields and the characteristic of the field. First notice that in fields of characteristic 2, the permanent is equal to the determinant and is therefore efficiently computable. Furthermore, there exists an elaborate yet polynomial time algorithm of Kogan [18] that computes the (orthogonal) matrix permanent over fields of characteristic 3. We prove a sharp dichotomy theorem: for fields of characteristic 2 or 3 there is an efficient procedure to compute orthogonal matrix permanents, and for all other primes there exists a finite field1 of characteristic p for which the permanent of an orthogonal matrix (over that field) is as hard as counting the number of solutions to a CNF formula mod p.2 Furthermore, there even exist infinitely many primes for which computing the permanent of an orthogonal matrix over Fp (i.e., modulo p) is hard. Finally, we give a polynomial interpolation argument showing that the permanent of a positive semidefinite matrix is #P-hard. This has an interesting consequence due a recent connection between matrix permanents and boson sampling experiments with thermal input states [10, 21]. In particular, the probability of detecting one photon on each of a set of modes, and zero photons on the remaining modes, is proportional to a positive semidefinite matrix dependent on the temperatures of the input thermal states. Our result implies that we cannot in general compute such output probabilities exactly despite the fact that an efficient classical sampling algorithm exists [21]. In conclusion, we show that the linear optical framework can help us prove new classical re- sults in computer science. Indeed, we can even use the formalism to prove structural results about the permanent (e.g., increasing the eigenvalues λ1,...,λn of a positive semidefinite matrix Udiag λ ,...,λ U cannot decrease its permanent) for which we currently have no application, { 1 n} † but which might later prove useful (see Appendix E for details). Proof Outline. The main result concerning the #P-hardness of real orthogonal permanents follows from three major steps: 1. Construct a quantum circuit (over qubits) with the following property: If you can compute the probability of measuring the all-zeros state after the circuit has been applied to the all- zeros state, then you could calculate some #P-hard quantity. We must modify the original construction of Aaronson [2], so that all the gates in the used in this construction are real. 2. Use a modified version of the Knill, Laflamme, Milburn protocol [17] to construct a linear optical circuit which simulates the qubit circuit we constructed in the previous step. In particular, we must ensure that the linear optical circuit starts and ends with one photon in every mode.