In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.

In this paper, we propose a quantum algorithm that supports a real-valued higher-order unconstrained binary optimization (HUBO) problem. This algorithm is based on the Grover adaptive search that originally supported HUBO with integer coefficients. Next, as an application example, we formulate multiple-input multiple-output maximum likelihood detection as a HUBO problem with real-valued coefficients, where we use the Gray-coded bit-to-symbol mapping specified in the 5G standard. The proposed approach allows us to construct an efficient quantum circuit for the detection problem and to analyze specific numbers of required qubits and quantum gates, whereas other conventional studies have assumed that such a circuit is feasible as a quantum oracle. To further accelerate the quantum algorithm, we also derive a probability distribution of the objective function value and determine a unique threshold to sample better states. Assuming a future fault-tolerant quantum computing, our proposed algorithm has the potential for significantly reducing query complexity in the classical domain and providing a quadratic speedup in the quantum domain.

Abstract The fastest known classical algorithm deciding the k-colorability of n-vertex graph requires running time $$\varOmega (2^n)$$ Ω ( 2 n ) for $$k\ge 5$$ k ≥ 5 . In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time $$O(1.9140^n)$$ O ( 1 . 9140 n ) using quantum random access memory (QRAM). Our approach is based on Ambainis et al’s quantum dynamic programming with applications of Grover’s search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 20-coloring problem with running time $$O(1.9575^n)$$ O ( 1 . 9575 n ) . For the polynomial-space quantum algorithm, we essentially develop $$(4-\epsilon )^n$$ ( 4 - ϵ ) n -time classical algorithms that can be improved quadratically by Grover’s search.

Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within$\exp(-\Omega(m\log m))$additive error is hard for any classical computer, where$m$is the number of gates in the quantum computation. More precisely, we show that the above problem is #P-hard under BPPNPreduction. In the recent experiments, the quantum circuit has n-qubits and the architecture is a two-dimensional grid of size$\sqrt{n}\times\sqrt{n}$[1]. Indeed for constant depth circuits approximating the output probabilities to within$2^{-\Omega(n\log n)}$is hard. For circuits of depth$\log n$or$\sqrt{n}$for which the anti-concentration property holds, approximating the output probabilities to within$2^{-\Omega(n\log^{2}n)}$and$2^{-\Omega(n^{3/2}\log n)}$is hard respectively. We then show that the hardness results extend to any open neighborhood of an arbitrary (fixed) circuit including the trivial circuit with identity gates. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp–Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.

Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the $n$-dimensional lattice graph $Q(D,n)$ with vertices in $\{0,1,\ldots,D\}^n$. We study the complexity of the following problem: given a subgraph $G$ of $Q(D,n)$ via query access to the edges, determine whether there is a path from $0^n$ to $D^n$. While the classical query complexity is $\widetildeΘ((D+1)^n)$, we show a quantum algorithm with complexity $\widetilde O(T_D^n)$, where $T_D < D+1$. The first few values of $T_D$ are $T_1 \approx 1.817$, $T_2 \approx 2.660$, $T_3 \approx 3.529$, $T_4 \approx 4.421$, $T_5 \approx 5.332$. We also prove that $T_D \geq \frac{D+1}{\mathrm e}$, thus for general $D$, this algorithm does not provide, for example, a speedup, polynomial in the size of the lattice. While the presented quantum algorithm is a natural generalization of the known quantum algorithm for $D=1$ by Ambainis et al., the analysis of complexity is rather complicated. For the precise analysis, we use the saddle-point method, which is a common tool in analytic combinatorics, but has not been widely used in this field. We then show an implementation of this algorithm with time complexity $\text{poly}(n)^{\log n} T_D^n$, and apply it to the Set Multicover problem. In this problem, $m$ subsets of $[n]$ are given, and the task is to find the smallest number of these subsets that cover each element of $[n]$ at least $D$ times. While the time complexity of the best known classical algorithm is $O(m(D+1)^n)$, the time complexity of our quantum algorithm is $\text{poly}(m,n)^{\log n} T_D^n$.

Let P:{0,1}k → {0,1} be a nontrivial k-ary predicate. Consider a random instance of the constraint satisfaction problem (P) on n variables with Δ n constraints, each being P applied to k randomly chosen literals. Provided the constraint density satisfies Δ ≫ 1, such an instance is unsatisfiable with high probability. The refutation problem is to efficiently find a proof of unsatisfiability.

A network topology with low average shortest path length (ASPL) provides efficient data transmission while the number of nodes and the number of links incident to each node are often limited due to physical constraints. In this paper, we consider the construction of low ASPL graphs under these constraints by using stochastic local search (SLS) algorithms. Since the ASPL cannot be calculated efficiently, the ASPL is not suitable for the evaluation function of SLS algorithms. We first derive an equality and bounds for the ASPL of graphs of diameter 3. On the basis of the simplest upper bound of the ASPL, we propose to use 3Δ + 2□ as the evaluation function for graphs of diameter 3 where Δ and □ denote the number of triangles and squares in a graph, respectively. We show that the proposed evaluation function can be evaluated in O(1) time as the number of nodes and the maximum degree tend to infinity by using some data tables. By using the simulated annealing with the proposed evaluation function, we construct low ASPL regular graphs of diameter 3 with 10 000 nodes.

The holographic transformation, belief propagation and loop calculus are generalized to problems in generalized probabilistic theories including quantum mechanics. In this work, the partition function of classical factor graph is represented by an inner product of two high-dimensional vectors both of which can be decomposed to tensor products of low-dimensional vectors. On the representation, the holographic transformation is clearly understood by using adjoint linear maps. Furthermore, on the formulation using inner product, the belief propagation is naturally defined from the derivation of the loop calculus formula. As a consequence, the holographic transformation, the belief propagation and the loop calculus are generalized to measurement problems in quantum mechanics and generalized probabilistic theories.

The Bethe approximation is a well-known approximation of the partition function used in statistical physics. Recently, an equality relating the partition function and its Bethe approximation was obtained for graphical models with binary variables by Chertkov and Chernyak. In this equality, the multiplicative error in the Bethe approximation is represented as a weighted sum over all generalized loops in the graphical model. In this paper, the equality is generalized to graphical models with nonbinary alphabet using concepts from information geometry.

Polarization phenomenon over any finite field Fq with size q being a power of a prime is considered. This problem is a generalization of the original proposal of channel polarization by Arıkan for the binary field, as well as its extension to a prime field by Sasoglu, Telatar, and Arıkan. In this paper, a necessary and sufficient condition of a matrix over a finite field Fqis shown under which any source and channel are polarized. Furthermore, the result of the speed of polarization for the binary alphabet obtained by Arıkan and Telatar is generalized to arbitrary finite field. It is also shown that the asymptotic error probability of polar codes is improved by using the Reed-Solomon matrices, which can be regarded as a natural generalization of the 2 × 2 binary matrix used in the original proposal by Arıkan.

We consider the asymptotic behavior of the polarization process in the large block-length regime when transmission takes place over a binary-input memoryless symmetric channel$W$. In particular, we study the asymptotics of the cumulative distribution$\BBP(Z_{n}\leq z)$, where$\{Z_{n}\}$is the Bhattacharyya process associated with$W$, and its dependence on the rate of transmission. On the basis of this result, we characterize the asymptotic behavior, as well as its dependence on the rate, of the block error probability of polar codes using the successive cancellation decoder. This refines the original asymptotic bounds by Arıkan and Telatar. Our results apply to general polar codes based on$\ell\times\ell$kernel matrices. We also provide asymptotic lower bounds on the block error probability of polar codes using the maximum a posteriori (MAP) decoder. The MAP lower bound and the successive cancellation upper bound coincide when$\ell=2$, but there is a gap for$\ell > 2$.

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For fixed numbers of BP iterations, the bit error probability approaches a limit as the blocklength tends to infinity, and the limit is obtained via density evolution. The finite-blocklength correction behaves like α(ε,t)/n+Θ(n-2) as the blocklengthntends to infinity where α(ε,t) denotes a specific constant determined by the code ensemble considered, the numbertof iterations, and the erasure probability ε of the BEC. In this paper, we derive a set of recursive formulas which allows the evaluation of the constant α(ε,t) for standard irregular ensembles. The dominant difference α(ε,t)/ncan be considered as effects of cycle-free and single-cycle structures of local graphs. Furthermore, it is confirmed via numerical simulations that estimation of the bit error probability using α(ε,t) is accurate even for small blocklengths.

In statistical physics and information theory, while asymptotic behavior of the partition function is often of our primary interest, the most of works are dedicated to analysis of the exponent of the partition function. In our previous paper on sparse random factor graph ensembles, we show that the exponent of the expectation of the partition function is represented as the minimum of the Bethe free energy of the small averaged graph by using the method of types. In this paper, we present a general framework to study more precise asymptotic behaviors of the partition function, using the central approximation in conjunction with the method of types.

Convolutional Low-Density-Parity-Check (LDPC) ensembles have excellent performance. Their iterative threshold increases with their average degree, or with the size of the coupling window in randomized constructions. In the latter case, as the window size grows, the Belief Propagation (BP) threshold attains the maximum-a-posteriori (MAP) threshold of the underlying ensemble. In this contribution we show that a similar phenomenon happens for the growth rate of coupled ensembles. Loosely speaking, we observe that as the coupling strength grows, the growth rate of the coupled ensemble comes close to the concave hull of the underlying ensemble's growth rate. For ensembles randomly coupled across a window the growth rate actually tends to the concave hull of the underlying one as the window size increases. Our observations are supported by the calculations of the combinatorial growth rate, and that of the growth rate derived from the replica method. The observed concavity is a general feature of coupled mean field graphical models and is already present at the level of coupled Curie-Weiss models. There, the canonical free energy of the coupled system tends to the concave hull of the underlying one. As we explain, the behavior of the growth rate of coupled ensembles is exactly analogous.

Recently, Vontobel showed the relationship between Bethe free energy and annealed free energy for protograph factor graph ensembles. In this paper, annealed free energy of any random regular factor graph ensembles are connected to Bethe free energy. The annealed free energy is expressed as the solution of maximization problem whose stationary condition coincides with equations of belief propagation since the contribution to partition function of particular type of variable and factor nodes has similar form of minus Bethe free energy. It gives simple derivation of quenched free energy by using the replica method. It implies equivalence of the replica and cavity methods for any random irregular factor graph ensembles. As consequence, it is shown that the replica symmetric solution and annealed free energy are equal for regular ensemble.

A method of channel polarization, proposed by Arikan, allows us to construct efficient capacity-achieving channel codes. In the original work, binary input discrete memoryless channels are considered. A special case of q-ary channel polarization is considered by Şaşoğlu, Telatar, and Arikan. In this paper, we consider more general channel polarization on q-ary channels. We further show explicit constructions using Reed-Solomon codes, on which asymptotically fast channel polarization is induced.

A rate-dependent upper bound of the best achievable block error probability of polar codes with successive-cancellation decoding is derived.

Polar codes, introduced by Arıkan, achieve symmetric capacity of any discrete memoryless channels under low encoding and decoding complexity. Recently, non-binary polar codes have been investigated. In this paper, we calculate error probability of non-binary polar codes constructed on the basis of Reed-Solomon matrices by numerical simulations. It is confirmed that 4-ary polar codes have significantly better performance than binary polar codes on binary-input AWGN channel. We also discuss an interpretation of polar codes in terms of algebraic geometry codes, and further show that polar codes using Hermitian codes have asymptotically good performance.

Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength n tends to infinity. The difference between the bit error probability with blocklength n and the large-blocklength limit behaves asymptotically like ¿/n, where the coefficient ¿ depends on the ensemble, the number of iterations and the erasure probability of the BEC. In, ¿ is calculated for regular ensembles. In this paper, ¿ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of ¿ to the analytic result is significantly fast for irregular unexpurgated ensembles.

Channel polarization is a method of constructing capacity achieving codes for symmetric binary-input discrete memoryless channels (B-DMCs). In the original paper, the construction complexity is exponential in the blocklength. In this paper, a new construction method for arbitrary symmetric binary memoryless channel (B-MC) with linear complexity in the blocklength is proposed. Furthermore, new upper bound and lower bound of the block error probability of polar codes are derived for the BEC and arbitrary symmetric B-MC, respectively.

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. Furthermore, a gap between the bit error probability after finite number of iterations for finite block length n and that for infinite block length is asymptotically α/n, where α denotes a speci..c constant determined by a degree distribution, a number of iterations and erasure probability. Our main result is to derive an ef..cient algorithm for calculating α for regular ensembles.