Permutational Quantum Computing (Jordan, 2009) is a natural quantum computational model that was conjectured to capture non-classical aspects of quantum computation. Contrary to the previous conjecture, we find a classical algorithm that efficiently approximates output distributions of the model up to polynomially small additive precision. We extend this algorithm to show that a large class of important quantum circuits - the Quantum Schur Sampling circuits - can be also efficiently simulated classically. The algorithm can be used to efficiently estimate non-trivial elements of irreducible representation matrices of the symmetric group on
n elements and might be also used to approximate transition amplitudes of the Ponzano-Regge model.
A large class of quantum algorithms can be represented by placing a reversible classical circuit, or diagonal gates, between a pair of quantum Fourier transformations. Notable examples include the circuit from Shor's factoring algorithm and IQP circuits. Here we study quantum circuits with a similar structure, where the QFT has been replaced by a sequentially coupled quantum Schur transform. We extend the Kushilevitz-Mansour algorithm for sampling from Fourier-sparse output distributions to these circuits. It is shown that samples from a classical distribution close to the output of a sparse Schur circuit can be efficiently generated by this classical method. In particular, it implies that circuits as found in a computationally interesting regime of Permutational Quantum Computing can be sampled from efficiently classically, assuming they give rise to sparse output distributions. Numerical simulations indicate that a significant fraction of Permutational circuits on a small number of qubits have this property, allowing for efficient sampling of their output.
Permutational quantum computing (PQC) [Quantum Inf. Comput., 10, 470-497 (2010)QICUAW1533-7146] is a natural quantum computational model conjectured to capture nonclassical aspects of quantum computation. An argument backing this conjecture was the observation that there was no efficient classical algorithm for estimation of matrix elements of the S_{n} irreducible representation matrices in the Young's orthogonal form, which correspond to transition amplitudes of a broad class of PQC circuits. This problem can be solved with a PQC machine in polynomial time, but no efficient classical algorithm for the problem was previously known. Here we give a classical algorithm that efficiently approximates the transition amplitudes up to polynomial additive precision and hence solves this problem. We further extend our discussion to show that transition amplitudes of a broader class of quantum circuits-the quantum Schur sampling circuits-can also be efficiently classically approximated.
How powerful are Quantum Computers? Despite the prevailing belief that Quantum Computers are more powerful than their classical counterparts, this remains a conjecture backed by little formal evidence. Shor's famous factoring algorithm [Shor97] gives an example of a problem that can be solved efficiently on a quantum computer with no known efficient classical algorithm. Factoring, however, is unlikely to be NP-Hard, meaning that few unexpected formal consequences would arise, should such a classical algorithm be discovered. Could it then be the case that any quantum algorithm can be simulated efficiently classically? Likewise, could it be the case that Quantum Computers can quickly solve problems much harder than factoring? If so, where does this power come from, and what classical computational resources do we need to solve the hardest problems for which there exist efficient quantum algorithms? We make progress toward understanding these questions through studying the relationship between classical nondeterminism and quantum computing. In particular, is there a problem that can be solved efficiently on a Quantum Computer that cannot be efficiently solved using nondeterminism? In this thesis we address this problem from the perspective of sampling problems. Namely, we give evidence that approximately sampling the Quantum Fourier Transform of an efficiently computable function, while easy quantumly, is hard for any classical machine in the Polynomial Time Hierarchy. In particular, we prove the existence of a class of distributions that can be sampled efficiently by a Quantum Computer, that likely cannot be approximately sampled in randomized polynomial time with an oracle for the Polynomial Time Hierarchy. Our work complements and generalizes the evidence given in Aaronson and Arkhipov's work [AA2013] where a different distribution with the same computational properties was given. Our result is more general than theirs, but requires a more powerful quantum sampler.
This thesis presents an efficient quantum algorithm and explicit circuits for generating eigenstates of arbitrary SU(2) and SU(3) representations. These include a wide variety of highly entangled states. The algorithm uses Schur transform that rotates the input computational basis states to the output Schur basis states with resources polynomial in number of qudits n. Using the fact that quantum logic is reversible, we accomplish the desired result using the inverse Schur transform. The algorithm can be easily generalized to any arbitrary higher groups.
One-parameter interpolations between any two unitary matrices (e.g., quantum gates)
U1 and
U2 along efficient paths contained in the unitary group are constructed. Motivated by applications, we propose the continuous unitary path
U(θ) obtained from the QR-factorization
U(θ)R(θ)=(1−θ)A+θB, where
U1R1=A and
U2R2=B are the QR-factorizations of
A and
B, and
U(θ) is a unitary for all
θ with
U(0)=U1 and
U(1)=U2. The QR-algorithm is modified to, instead of
U(θ), output a matrix whose columns are proportional to the corresponding columns of
U(θ) and whose entries are polynomial or rational functions of
θ. By an extension of the Berlekamp-Welch algorithm we show that rational functions can be efficiently and exactly interpolated with respect to
θ. We then construct probability distributions over unitaries that are arbitrarily close to the Haar measure. Demonstration of computational advantages of NISQ over classical computers is an imperative near-term goal, especially with the exuberant experimental frontier in academia and industry (e.g., IBM and Google). A candidate for quantum computational supremacy is Random Circuit Sampling (RCS), which is the task of sampling from the output distribution of a random circuit. The aforementioned mathematical results provide a new way of scrambling quantum circuits and are applied to prove that exact RCS is
#P-Hard on average, which is a simpler alternative to Bouland et al's. (Dis)Proving the quantum supremacy conjecture requires "approximate" average case hardness; this remains an open problem for all quantum supremacy proposals.