I have been thinking about ways to construct interesting adiabatic algorithms for tasks other than optimization. One of the areas I have been looking is in quantum simulation and in particular in calculating energy eigenvalues & eigenvectors of quantum systems.

Here is a question I was looking at earlier today that I don’t know the answer to, maybe you can help. Here is my question.

Assume you are given two Hermitian matrices and . What are the resource costs for determining if all the eigenvalues of and are the same, ie. they both have the same spectrum? Note that we don’t need to know what the eigenvalues actually are, just whether the two matrices have the same eigenvalues. One way to do it is to (a) diagonalize and , (b) sort each list of eigenvalues, and (c) compare them one by one. This is exponential in . I’m sure there must be a much better way to do this. Anyone have any helpful hints?

Here is a thought. If there exists an invertible matrix such that then and have the same eigenvalues. I think the phrase commonly used for these are similar matrices. So given two matrices is there an efficient way to check whether such a exists?

What about using the Smith normal form? I’m not sure what the efficiency of this method is, but basically two matrices, A and B, are similar if and only if the characteristic matrices, A-Ix and B-Ix (or Ix-A and Ix-B), have the same Smith normal form. The only hitch is that it assumes you’re working with a principle ideal domain (PID).

I’m not sure how hard it is, but it’s interesting that if you restrict P to being a permutation matrix, it’s exactly the Graph Isomorphism problem, which is very probably a much harder problem.

@quantummoxie: Nice attention to detail; luckily, it’s pretty likely he’ll be using a principle ideal domain, as that includes R, C, Z, and Z_2.🙂

Neil: Are you sure that if P is a permutation matrix this is graph isomorphism?

quantummoxie: yes the intent is that these are Hamiltonians (entries are complex).

“Neil: Are you sure that if P is a permutation matrix this is graph isomorphism?”

Looks that way to me. H1 and H2 are matrix representations of your two graphs and P is the vertex relabeling that gets you from one to the other.

Actually, I thought about it a bit more and it might not be graph isomorphism; if it was just one P instead of two, it would definitely be GI on a graph of 2^n nodes, since that’s exactly how one would define GI on a node-weighted, edge-weighted graph.

A lower bound on the time it’ll take is n(2^n), since even if you have diagonal matrices, you have to decide between (2^n)! orderings of the elements, which takes at least log((2^n)!) decisions, which scales as n(2^n). (Same argument as that sorting n elements takes nlogn time.) If the matrices are dense, a lower bound is 2^2n time, since there are 2^2n elements that affect the eigenvalues.

These correspond to nlogn and n^2 times for nxn matrices. I think diagonalization takes at least n^3 time for dense nxn matrices; maybe less for sparse, but I don’t know.

Hi Neil, yes… but I wonder if there is a way to know whether two matrices have the same eigenvalues without having to explicitly calculate them. Any scheme where you have to explicitly calculate all of them is a non-starter because of the exponential size of the matrices (say typically 2^1000 x 2^1000 sizes for a moderate quantum system).

Just matrix multiplication alone is exponentially expensive here. I can’t imagine solving this problem in less operations than a single multiplication. Do you have some polynomial time way to evaluate matrix-vector multiplications?

Jordan: No. But you don’t always have to explicitly work out all the elements of a matrix to know something interesting about it. For example the type of matrix we deal with here is just where is the jth Pauli matrix. This matrix has an exponential number of matrix elements but I can easily write it down in a compact form. For this problem I wonder if you can somehow answer the question just by looking at the “generators” of the matrix (in the case above the matrices).

I should also emphasize that what I’m looking for isn’t a calculation of the spectra, which obviously would be exponential. I’m wondering if there might be an efficient “oracle” that given two matrices can answer the decision problem “Are these two matrices similar?”

Geordie,

In the Smith normal form method I mentioned above, you don’t actually need to go through the step of calculating the eigenvalues. True, you need the characteristic matrix, which is not too many steps away, but it still saves a bit of calculating (I’m figuring calculating the Smith normal form is about as resource intensive as calculating the Jordan form which you’d have to do to diagonalize).

Ian

I think the problem is hard, at least the way you’ve phrased it so far. I assume that as per comment 7 that the way you really want to state this is that you have polynomial sized representations of the matrices H1 or H2….

Let f1 and f2 be two boolean functions which we specify by circuits in CNF form. Its then easy to construct a H1 such that H1=\sum_x f1(x)|x><x| but which has an efficient polynomial description…i.e. we can write H1 as a product of sums of Pauli Z's acting on the relevant variables. Same goes for H2. In this case your problem is the same as the circuit equivalence problem. But circuit equivalence is coNP-complete. So your problem is coNP-hard but it could be harder.

One way that it might not be so hard and which I think you might also be implicitly needing in a quantum algorithm for this is that H be a sum of a polynomial number of local terms. The H1 and H2 I constructed above does not have this property even though it has an efficient polynomial sized description. [(a+b)^10 is a small description of a term that is a sum of 2^10 terms]. Note that the representation form can have a big different. Testing circuit equivalence of binary decision diagrams is easy, for example.

So I think you need to define what you allow H1 and H2 to be given as better…

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