14 research outputs found
Processing Succinct Matrices and Vectors
We study the complexity of algorithmic problems for matrices that are
represented by multi-terminal decision diagrams (MTDD). These are a variant of
ordered decision diagrams, where the terminal nodes are labeled with arbitrary
elements of a semiring (instead of 0 and 1). A simple example shows that the
product of two MTDD-represented matrices cannot be represented by an MTDD of
polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by
allowing componentwise symbolic addition of variables (of the same dimension)
in rules. It is shown that accessing an entry, equality checking, matrix
multiplication, and other basic matrix operations can be solved in polynomial
time for MTDD_+-represented matrices. On the other hand, testing whether the
determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the
same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing
a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of
CSR 201
Boundaries of quantum supremacy via random circuit sampling
Google's recent quantum supremacy experiment heralded a transition point
where quantum computing performed a computational task, random circuit
sampling, that is beyond the practical reach of modern supercomputers. We
examine the constraints of the observed quantum runtime advantage in an
analytical extrapolation to circuits with a larger number of qubits and gates.
Due to the exponential decrease of the experimental fidelity with the number of
qubits and gates, we demonstrate for current fidelities a theoretical classical
runtime advantage for circuits beyond a depth of 100, while quantum runtimes
for cross-entropy benchmarking limit the region of a quantum advantage to
around 300 qubits. However, the quantum runtime advantage boundary grows
exponentially with reduced error rates, and our work highlights the importance
of continued progress along this line. Extrapolations of measured error rates
suggest that the limiting circuit size for which a computationally feasible
quantum runtime advantage in cross-entropy benchmarking can be achieved
approximately coincides with expectations for early implementations of the
surface code and other quantum error correction methods. Thus the boundaries of
quantum supremacy via random circuit sampling may fortuitously coincide with
the advent of scalable, error corrected quantum computing in the near term.Comment: 8 pages, 3 figure
An introduction to graphical tensor notation for mechanistic interpretability
Graphical tensor notation is a simple way of denoting linear operations on
tensors, originating from physics. Modern deep learning consists almost
entirely of operations on or between tensors, so easily understanding tensor
operations is quite important for understanding these systems. This is
especially true when attempting to reverse-engineer the algorithms learned by a
neural network in order to understand its behavior: a field known as
mechanistic interpretability. It's often easy to get confused about which
operations are happening between tensors and lose sight of the overall
structure, but graphical tensor notation makes it easier to parse things at a
glance and see interesting equivalences. The first half of this document
introduces the notation and applies it to some decompositions (SVD, CP, Tucker,
and tensor network decompositions), while the second half applies it to some
existing some foundational approaches for mechanistically understanding
language models, loosely following ``A Mathematical Framework for Transformer
Circuits'', then constructing an example ``induction head'' circuit in
graphical tensor notation.Comment: 30 pages, 75 figure
Classifying Complexity with the ZX-Calculus: Jones Polynomials and Potts Partition Functions
The ZX-calculus is a graphical language which allows for reasoning about
suitably represented tensor networks - namely ZX-diagrams - in terms of rewrite
rules. Here, we focus on problems which amount to exactly computing a scalar
encoded as a closed tensor network. In general, such problems are #P-hard.
However, there are families of such problems which are known to be in P when
the dimension is below a certain value. By expressing problem instances from
these families as ZX-diagrams, we see that the easy instances belong to the
stabilizer fragment of the ZX-calculus. Building on previous work on efficient
simplification of qubit stabilizer diagrams, we present simplifying rewrites
for the case of qutrits, which are of independent interest in the field of
quantum circuit optimisation. Finally, we look at the specific examples of
evaluating the Jones polynomial and of counting graph-colourings. Our
exposition further champions the ZX-calculus as a suitable and unifying
language for studying the complexity of a broad range of classical and quantum
problems.Comment: QPL 2021 submissio
Picturing counting reductions with the ZH-calculus
Counting the solutions to Boolean formulae defines the problem #SAT, which is
complete for the complexity class #P. We use the ZH-calculus, a universal and
complete graphical language for linear maps which naturally encodes counting
problems in terms of diagrams, to give graphical reductions from #SAT to
several related counting problems. Some of these graphical reductions, like to
#2SAT, are substantially simpler than known reductions via the matrix
permanent. Additionally, our approach allows us to consider the case of
counting solutions modulo an integer on equal footing. Finally, since the
ZH-calculus was originally introduced to reason about quantum computing, we
show that the problem of evaluating ZH-diagrams in the fragment corresponding
to the Clifford+T gateset, is in . Our results show that graphical
calculi represent an intuitive and useful framework for reasoning about
counting problems
A common algebraic description for probabilistic and quantum computations
AbstractThrough the study of gate arrays we develop a unified framework to deal with probabilistic and quantum computations, where the former is shown to be a natural special case of the latter. On this basis we show how to encode a probabilistic or quantum gate array into a sum-free tensor formula which satisfies the conditions of the partial trace problem, and vice-versa; that is, given a tensor formula F of order n脳1 over a semiring S plus a positive integer k, deciding whether the kth partial trace of the matrix valSn,n(F路FT) fulfills a certain property. We use this to show that a certain promise version of the sum-free partial trace problem is complete for the class pr- BPP (promise BPP) for formulas over the semiring (Q+,+,路) of the positive rational numbers, for pr-BQP (promise BQP) in the case of formulas defined over the field (Q+,+,路), and if the promise is given up, then completeness for PP is shown, regardless whether tensor formulas over positive rationals or rationals in general are used. This suggests that the difference between probabilistic and quantum polytime computers may ultimately lie in the possibility, in the latter case, of having destructive interference between computations occurring in parallel. Moreover, by considering variants of this problem, classes like 鈯昉, NP, C=P, its complement co-C=P, the promise version of Valiant's class UP, its generalization promise SPP, and unique polytime US can be characterized by carrying the problem properties and the underlying semiring