arXiv subjects

Pawel Wocjan

Publications and source records attributed to Pawel Wocjan.

At least 73 records · Page 4Linked to original sources

arXiv2006

Several natural BQP-Complete problems

A central problem in quantum computing is to identify computational tasks which can be solved substantially faster on a quantum computer than on any classical computer. By studying the hardest such tasks, known as BQP-complete problems, we deepen our understanding of the power and limitations of quantum computers. We present several BQP-complete problems, including Local Hamiltonian Eigenvalue Sampling and Phase Estimation Sampling. Different than the previous known BQP-complete problems (the Quadratically Signed Weight Enumerator problem [KL01] and the Approximation of Jones Polynomials [FKW02, FLW02, AJL06]), our problems are of a basic linear algebra nature and are closely related to the well-known quantum algorithm and quantum complexity theories.

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arXiv2005

On the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems

We consider an approach to deciding isomorphism of rigid n-vertex graphs (and related isomorphism problems) by solving a nonabelian hidden shift problem on a quantum computer using the standard method. Such an approach is arguably more natural than viewing the problem as a hidden subgroup problem. We prove that the hidden shift approach to rigid graph isomorphism is hard in two senses. First, we prove that Omega(n) copies of the hidden shift states are necessary to solve the problem (whereas O(n log n) copies are sufficient). Second, we prove that if one is restricted to single-register measurements, an exponential number of hidden shift states are required.

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arXiv2005

Real Mutually Unbiased Bases

We tabulate bounds on the optimal number of mutually unbiased bases in R^d. For most dimensions d, it can be shown with relatively simple methods that either there are no real orthonormal bases that are mutually unbiased or the optimal number is at most either 2 or 3. We discuss the limitations of these methods when applied to all dimensions, shedding some light on the difficulty of obtaining tight bounds for the remaining dimensions that have the form d=16n^2, where n can be any number. We additionally give a simpler, alternative proof that there can be at most d/2+1 real mutually unbiased bases in dimension d instead of invoking the known results on extremal Euclidean line sets by Cameron and Seidel, Delsarte, and Calderbank et al.

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arXiv2005

On independent permutation separability criteria

Recently P. Wocjan and M. Horodecki [quant-ph/0503129] gave a characterization of combinatorially independent permutation separability criteria. Combinatorial independence is a necessary condition for permutations to yield truly independent criteria meaning that that no criterion is strictly stronger that any other. In this paper we observe that some of these criteria are still dependent and analyze why these dependencies occur. To remove them we introduce an improved necessary condition and give a complete classification of the remaining permutations. We conjecture that the remaining class of criteria only contains truly independent permutation separability criteria. Our conjecture is based on the proof that for two, three and four parties all these criteria are truly independent and on numerical verification of their independence for up to 8 parties. It was commonly believed that for three parties there were 9 independent criteria, here we prove that there are exactly 6 independent criteria for three parties and 22 for four parties.

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arXiv2005

Characterization of combinatorially independent permutation separability criteria

The so-called permutation separability criteria are simple operational conditions that are necessary for separability of mixed states of multipartite systems: (1) permute the indices of the density matrix and (2) check if the trace norm of at least one of the resulting operators is greater than one. If it is greater than one then the state is necessarily entangled. A shortcoming of the permutation separability criteria is that many permutations give rise to dependent separability criteria. Therefore, we introduce a necessary condition for two permutations to yield independent criteria called combinatorical independence. This condition basically means that the map corresponding to one permutation cannot be obtained by concatenating the map corresponding to the second permutation with a norm-preserving map. We characterize completely combinatorically independent criteria, and determine simple permutations that represent all independent criteria. The representatives can be visualized by means of a simple graphical notation. They are composed of three basic operations: partial transpose, and two types of so-called reshufflings. In particular, for a four-partite system all criteria except one are composed of partial transpose and only one type of reshuffling; the exceptional one requires the second type of reshuffling. Furthermore, we show how to obtain efficiently for every permutation a simple representative. This method allows to check easily if two permutations are combinatorically equivalent or not.

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arXiv2005

Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.

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arXiv2004

The limitations of nice mutually unbiased bases

Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches. We prove this by establishing a correspondence between nice mutually unbiased bases and abelian subgroups of the index group of a nice error basis and then bounding the number of such subgroups. This bound also has implications for the construction of certain combinatorial objects called nets.

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arXiv2004

Efficient decoupling schemes with bounded controls based on Eulerian orthogonal arrays

The task of decoupling, i.e., removing unwanted interactions in a system Hamiltonian and/or couplings with an environment (decoherence), plays an important role in controlling quantum systems. There are many efficient decoupling schemes based on combinatorial concepts like orthogonal arrays, difference schemes and Hadamard matrices. So far these (combinatorial) decoupling schemes have relied on the ability to effect sequences of instantaneous, arbitrarily strong control Hamiltonians (bang-bang controls). To overcome the shortcomings of bang-bang control Viola and Knill proposed a method called Eulerian decoupling that allows the use of bounded-strength controls for decoupling. However, their method was not directly designed to take advantage of the composite structure of multipartite quantum systems. In this paper we define a combinatorial structure called an Eulerian orthogonal array. It merges the desirable properties of orthogonal arrays and Eulerian cycles in Cayley graphs (that are the basis of Eulerian decoupling). We show that this structure gives rise to decoupling schemes with bounded-strength control Hamiltonians that can be applied to composite quantum systems with few body Hamiltonians and special couplings with the environment. Furthermore, we show how to construct Eulerian orthogonal arrays having good parameters in order to obtain efficient decoupling schemes.

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arXiv2004

Equivalence of Decoupling Schemes and Orthogonal Arrays

We consider the problem of switching off unwanted interactions in a given multi-partite Hamiltonian. This is known to be an important primitive in quantum information processing and several schemes have been presented in the literature to achieve this task. A method to construct decoupling schemes for quantum systems of pairwise interacting qubits was introduced by M. Stollsteimer and G. Mahler and is based on orthogonal arrays. Another approach based on triples of Hadamard matrices that are closed under pointwise multiplication was proposed by D. Leung. In this paper, we show that both methods lead to the same class of decoupling schemes. Moreover, we establish a characterization of orthogonal arrays by showing that they are equivalent to decoupling schemes which allow a refinement into equidistant time-slots. Furthermore, we show that decoupling schemes for networks of higher-dimensional quantum systems with t-local Hamiltonians can be constructed from classical error-correcting codes.

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arXiv2004

New Construction of Mutually Unbiased Bases in Square Dimensions

We show that k=w+2 mutually unbiased bases can be constructed in any square dimension d=s^2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines the design-theoretic objects (k,s)-nets (which can be constructed from w mutually orthogonal Latin squares of order s and vice versa) and generalized Hadamard matrices of size s. Using known lower bounds on the asymptotic growth of the number of mutually orthogonal Latin squares (based on number theoretic sieving techniques), we obtain that the number of mutually unbiased bases in dimensions d=s^2 is greater than s^1/14.8 for all s but finitely many exceptions. Furthermore, our construction gives more mutually orthogonal bases in many non-prime-power dimensions than the construction that reduces the problem to prime power dimensions.

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arXiv2004

Ergodic quantum computing

We propose a (theoretical ;-) model for quantum computation where the result can be read out from the time average of the Hamiltonian dynamics of a 2-dimensional crystal on a cylinder. The Hamiltonian is a spatially local interaction among Wigner-Seitz cells containing 6 qubits. The quantum circuit that is simulated is specified by the initialization of program qubits. As in Margolus' Hamiltonian cellular automaton (implementing classical circuits), a propagating wave in a clock register controls asynchronously the application of the gates. However, in our approach all required initializations are basis states. After a while the synchronizing wave is essentially spread around the whole crystal. The circuit is designed such that the result is available with probability about 1/4 despite of the completely undefined computation step. This model reduces quantum computing to preparing basis states for some qubits, waiting, and measuring in the computational basis. Even though it may be unlikely to find our specific Hamiltonian in real solids, it is possible that also more natural interactions allow ergodic quantum computing.

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arXiv2004

Estimating mixing properties of local Hamiltonian dynamics and continuous quantum random walks is PSPACE-hard

A major topic of (classical) ergodic theory is to examine qualitatively how the phase space of dynamical systems is penetrated by the orbits of their dynamics. We consider interacting qubit systems with dynamics according to 4-local Hamiltonians and continuous quantum random walks. For these systems one could use the von Neumann entropy of the time-average to characterize the mixing properties of the corresponding orbits, i.e., what portion of the state space and how uniformly it is filled out by the orbits. We show that the problem of estimating this entropy is PSPACE-hard.

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arXiv2003

Measuring 4-local n-qubit observables could probabilistically solve PSPACE

We consider a hypothetical apparatus that implements measurements for arbitrary 4-local quantum observables A on n qubits. The apparatus implements the ``measurement algorithm'' after receiving a classical description of A. We show that a few precise measurements, applied to a basis state would provide a probabilistic solution of PSPACE problems. The error probability decreases exponentially with the number of runs if the measurement accuracy is of the order of the spectral gaps of A. Moreover, every decision problem which can be solved on a quantum computer in T time steps can be encoded into a 4-local observable such that the solution requires only measurements of accuracy O(1/T). Provided that BQP<>PSPACE, our result shows that efficient algorithms for precise measurements of general 4-local observables cannot exist. We conjecture that the class of physically existing interactions is large enough to allow the conclusion that precise energy measurements for general many-particle systems require control algorithms with high complexity.

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arXiv2003

Two QCMA-complete problems

QMA and QCMA are possible quantum analogues of the complexity class NP. In QCMA the verifier is a quantum program and the proof is classical. In contrast, in QMA the proof is also a quantum state. We show that two known QMA-complete problems can be modified to QCMA-complete problems in a natural way: (1) Deciding whether a 3-local Hamiltonian has low energy states (with energy smaller than a given value) that can be prepared with at most k elementary gates is QCMA-complete, whereas it is QMA-complete when the restriction on the complexity of preparation is dropped. (2) Deciding whether a (classically described) quantum circuit acts almost as the identity on all basis states is QCMA-complete. It is QMA-complete to decide whether it acts on all states almost as the identity.

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arXiv2003

Identity check is QMA-complete

We define the problem identity check: Given a classical description of a quantum circuit, determine whether it is almost equivalent to the identity. Explicitly, the task is to decide whether the corresponding unitary is close to a complex multiple of the identity matrix with respect to the operator norm. We show that this problem is QMA-complete. A generalization of this problem is equivalence check: Given two descriptions of quantum circuits and a description of a common invariant subspace, decide whether the restrictions of the circuits to this subspace almost coincide. We show that equivalence check is also in QMA and hence QMA-complete.

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arXiv2003

Cooling and Low Energy State Preparation for 3-local Hamiltonians are FQMA-complete

We introduce the quantum complexity class FQMA. This class describes the complexity of generating a quantum state that serves as a witness for a given QMA problem. In a certain sense, FQMA is the quantum analogue of FNP (function problems associated with NP). The latter describes the complexity of finding a succinct proof for a NP decision problem. Whereas all FNP problems can be reduced to NP, there is no obvious reduction of FQMA to QMA since the solution of FQMA is a quantum state and the solution of QMA the answer yes or no. We consider quantum state generators that get classical descriptions of 3-local Hamiltonians on n qubits as input and prepare low energy states for these systems as output. We show that such state generators can be used to prepare witnesses for QMA problems. Hence low energy state preparation is FQMA-complete. Our proofs are extensions of the proofs by Kitaev et al. and Kempe and Regev for the QMA-completeness of k-local Hamiltonian problems. We show that FQMA can be solved by preparing thermal equilibrium states with an appropriate temperature decreasing as the reciprocal of a polynomial in n.

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arXiv2003

Treating the Independent Set Problem by 2D Ising Interactions with Adiabatic Quantum Computing

We construct a nearest-neighbor Hamiltonian whose ground states encode the solutions to the NP-complete problem INDEPENDENT SET in cubic planar graphs. The Hamiltonian can be easily simulated by Ising interactions between adjacent particles on a 2D rectangular lattice. We describe the required pulse sequences. Our methods could help to implement adiabatic quantum computing by physically reasonable Hamiltonians like short-range interactions.

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arXiv2003

The 2-local Hamiltonian problem encompasses NP

We show that the NP complete problems MAX CUT and INDEPENDENT SET can be formulated as the 2-local Hamiltonian problem as defined by Kitaev. He introduced the quantum complexity class BQNP as the quantum analog of NP, and showed that the 5-local Hamiltonian problem is BQNP-complete. It is not known whether the s-local Hamiltonian problem is BQNP-complete for s smaller than 5. Therefore it is interesting to determine what problems can be reduced to the s-local Hamiltonian problem. Kitaev showed that 3-SAT can be formulated as a 3-local Hamiltonian problem. We extend his result by showing that 2-locality is sufficient in order to encompass NP.

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