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We study sheaf codes, a type of linear codes with a fixed hierarchical collection of local codes, viewed as a sheaf of vector spaces on a finite topological space we call coded space. Many existing codes, such as tensor product codes, Sipser-Spielman codes, and their more recent high-dimensional analogs, can be naturally represented as sheaf codes on simplicial and cubical complexes, considered as coded spaces. We introduce a new property of a sheaf code, called maximal extendibility, which ensures that within a class of codes on the same coded space, we encounter as few obstructions as possible when extending local sections globally. We show that in every class of sheaf codes defined on the same space and parameterized by parity-check matrices with polynomial entries, there always exists a maximally extendable sheaf code. Such codes are very interesting since it is possible to show that maximally extendable tensor product codes are good coboundary expanders, which potentially could be used to attack the qLTC conjecture., Comment: 17 pages

We study the coboundary expansion property of product codes called product expansion, which played a key role in all recent constructions of good qLDPC codes. It was shown before that this property is equivalent to robust testability and agreement testability for products of two codes with linear distance. First, we show that robust testability for product of many codes with linear distance is equivalent to agreement testability. Second, we provide an example of product of three codes with linear distance which is robustly testable but not product expanding., Comment: 11 pages

A new version of quantum hashing technique is developed wherein a quantum hash is constructed as a sequence of single-photon high-dimensional states (qudits). A proof-of-principle implementation of the high-dimensional quantum hashing protocol using orbital-angular momentum encoding of single photons is implemented. It is shown that the number of qudits decreases with increase of their dimension for an optimal ratio between collision probability and decoding probability of the hash. Thus, increasing dimension of information carriers makes quantum hashing with single photons more efficient.

We present results on experimental studies of spin current, measured under spin pumping at ferromag-netic resonance in wide frequency band 2-20 GHz for SrIrO3/La0.7Sr0.3MnO3 heterostructures fabricated by RF magnetron sputtering at high temperature. The epitaxial growth of the thin film in heterostructure by a cube-on-cube mechanism was confirmed by XRD and TEM analysis. Taking into account the con-tribution of anisotropic magnetoresistance the spin current was estimated as 1/3 of the total response. We show that both real and imaginary parts of spin mixing conductance are valuable for heterostructures with strong spin-orbit interaction in SrIrO3. Imaginary part of spin mixing conductance was estimated by means of shift of ferromagnetic resonance field of La0.7Sr0.3MnO3 layer in heterostructure. The spin mag-netoresistance was evaluated from angular dependencies of magnetoresistance measured in planar Hall configuration. In order to extract the influence of anisotropic magnetoresistance a La0.7Sr0.3MnO3 film was measured as well. The spin Hall angle for heterostructure was found higher than for interface Pt/ La0.7Sr0.3MnO3., Comment: 21 pages, 11 figures

We show that the tensor product of two random linear codes is robustly testable with high probability. This implies that one can obtain pairs of linear codes such that their product and the product of their dual codes are simultaneously robustly testable. Such two-sided robustly testable codes (with a much weaker form of robustness) were the key ingredient in the recent constructions of asymptotically good quantum LDPC codes, which ensured their linear minimum distance. We hope that the existence of such codes with a stronger form of robustness, shown here, can be used to simplify the proofs and provide better distance bounds in these constructions. We also give new very simple examples of non-robustly testable codes. We show that if the parity-checks of two codes are mutually orthogonal, then their product is not robustly testable. In particular, this implies that the product of a code with its dual can never be robustly testable. We also study a property of a collection of linear codes called product-expansion, which can be viewed as a coboundary expansion of the cochain complex naturally associated with the product of these codes. We show that this property is related with the robust testability and the agreement testability of the products of codes., Comment: 26 pages, 3 figures

The term "Cleaning Lemma" refers to a family of similar propositions that have been used in Quantum Coding Theory to estimate the minimum distance of a code in terms of its length and dimension. We show that the mathematical core is a simple fact of linear algebra of inner product spaces; moreover, it admits a further reduction to a combinatorial, lattice-theoretical level. Several concrete variants of the Cleaning Lemma and some additional propositions are derived as corollaries of the proposed approach., Comment: 25pp

Random circuit sampling has become a popular means for demonstrating the superiority of quantum computers over classical supercomputers. While quantum chips are evolving rapidly, classical sampling algorithms are also getting better and better. The major challenge is to generate bitstrings exhibiting an XEB fidelity above that of the quantum chips. Here we present a classical sampling algorithm for producing the probability distribution of any given random quantum circuit, where the fidelity can be rigorously bounded. Specifically, our algorithm performs rejection sampling after the introduced very recently multi-tensor contraction algorithm. We show that the fidelity can be controlled by partially contracting the dominant paths in the tensor network and by adjusting the number of batches used in the rejection sampling. As a demonstration, we classically produced 1 million samples with the fidelity bounded by 0.2%, based on the 20-cycle circuit of the Sycamore 53-qubit quantum chip. Though this task was initially estimated to take 10,000 years on the Summit supercomputer, it took about 14.5 days using our algorithm on a relatively small cluster with 32 GPUs (Tesla V100 16GB). Furthermore, we estimate that for the Zuchongzhi 56-qubit 20-cycle circuit one can produce 1M samples with fidelity 0.066% using the Selene supercomputer with 4480 GPUs (Tesla A100 80GB) in about 4 days., Comment: 10 pages, 4 figures, 4 tables

We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups. We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. Moreover, we show that the produced classical LDPC codes are also asymptotically good and locally testable with constant query and soundness parameters, which proves a well-known conjecture in the field of locally testable codes., Comment: Updated the introduction, corrected some misprints, clarified some proofs, added some new bibliography including arXiv:2005.01045 containing an independent construction of good LTCs

Quantum hashing is a promising generalization of the cryptographic hashing concept on the quantum domain. In this paper, we construct a quantum hash via a sequence of single-photon states and perform a proof-of-principle experiment using orbital angular momentum (OAM) encoding. We experimentally verify the collision resistance of the quantum hash function depending on the number of qubits in use. Based on these results, we conclude that theoretical estimates are confirmed for different bases of OAM states and the proposed technique can be useful in computational and cryptographic scenarios. The possibility of multiplexing different OAM bases can make this approach even more efficient., Comment: Accepted for publication in PRA

The computational advantage of noisy quantum computers has been demonstrated by sampling the bitstrings of quantum random circuits. An important issue is how the performance of quantum devices could be quantified in the so-called "supremacy regime". The standard approach is through the linear cross entropy benchmark (XEB), where the theoretical value of the probability is required for each bitstring. However, the computational cost of XEB grows exponentially. So far, random circuits of the 53-qubit Sycamore chip were verified up to 10 cycles of gates only; the XEB fidelities of deeper circuits were approximated with simplified circuits instead. Here we present a multi-tensor contraction algorithm for speeding up the calculations of XEB for quantum circuits, where the computational cost can be significantly reduced through some form of memoization. As a demonstration, we analyzed the experimental data of the 53-qubit Sycamore chip and obtained the exact values of the corresponding XEB fidelities up to 16 cycles using only moderate computing resources (few GPUs). If the algorithm was implemented on the Summit supercomputer, we estimate that for the supremacy (20 cycles) circuits, it would only cost 7.5 days, which is several orders of magnitude lower than previously estimated in the literature., Comment: 13 pages, 8 figures

In this paper we research a model of multilayer circuits with a single logical layer. We consider $\lambda$-separable graphs as a support for circuits. We establish the Shannon function lower bound $\max \bigl(\frac{2^n}{n}, \frac{2^n (1 - \lambda)}{\log k} \bigr)$ for this type of circuits where $k$ is the number of layers. For $d$-dimensional graphs, which are $\lambda$-separable for $\lambda = \frac{d - 1}{d}$, this gives the Shannon function lower bound $\frac{2^n}{\min(n, d \log k)}$. For multidimensional rectangular circuits the proved lower bound asymptotically matches to the upper bound.

We give a construction of quantum LDPC codes of dimension $\Theta(\log N)$ and distance $\Theta(N/\log N)$ as the code length $N\to\infty$. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance $\Omega(N^{1-\alpha/2}/\log N)$ and dimension $\Omega(N^\alpha \log N)$, where $0 \le \alpha < 1$. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed $R < 1$ there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least $R$ with, in some sense, optimal circulant size $\Omega(N/\log N)$ as the code length $N\to\infty$., Comment: 17 pages, 2 figures. Accepted for publication in IEEE Transactions on Information Theory

Electromagnetically induced transparency allows for controllable change of absorption properties which can be exploited in a number of applications including optical quantum memory. In this paper, we present a study of the electromagnetically induced transparency in $^{167}$Er:$^6$LiYF$_4$ crystal at low magnetic fields and ultra-low temperatures. Experimental measurement scheme employs optical vector network analysis which provides high precision measurement of amplitude, phase and pulse delay. We found that sub-Kelvin temperatures are the necessary requirement for studying electromagnetically induced transparency in this crystal at low fields. A good agreement between theory and experiment is achieved taking into account the phonon bottleneck effect., Comment: 8 pages, 5 figures

We study the generation of correlated photon pairs and heralded single photons via strongly non-degenerate spontaneous four-wave mixing (SFWM) in a series of identical micro-/nano fibers (MNF). Joint spectral intensity of the biphoton field generated at the wavelength of about 880 nm and 1310 nm has been measured under excitation by 100 ps laser pulses demonstrating good agreement with the theoretical prediction. The measured zero-time second-order autocorrelation function was about 0.2 when the emission rate of the heralded photons was of 4 Hz. The MNF-based source perfectly matches standard single-mode fibers, which makes it compatible with the existing fiber communication networks. In addition, SFWM observation in a series of identical MNFs allows increasing generation rate of single photons via spatial multiplexing., Comment: 7 pages, 11 figures

Er$^{3+}$:Y$_2$SiO$_5$ crystal is a promising candidate with a great variety of its potential applications in quantum information processing and quantum communications ranging from optical/microwave quantum memories to circuit QED and microwave-to-optics frequency converters. Some of the above listed applications require ultra-low temperature environment, i.e., temperatures $T\lesssim0.1~$K. Most of the experiments with erbium doped crystals have been so far carried out at temperatures above 1.5~K. Therefore, only little information is known about Er$^{3+}$:Y$_2$SiO$_5$ coherence properties at millikelvins. Here, we investigate optical decoherence of $^{167}$Er:Y$_2$SiO$_5$ crystal by performing 2- and 3-pulse echo experiments at millikelvin temperature range and at weak and moderate magnetic fields. We show that the deep freezing of the crystal results in an increase of optical coherence time by one order of magnitude compared to temperature of 1.5 Kelvin and magnetic field of $\sim$0.2~T, taken as a reference point. We further describe the detailed investigation of the decoherence mechanisms in this regime., Comment: 9 pages, 7 figures, 1 table, 47 references

A scheme for heralded generation of frequency-bin photonic qubits via spontaneous four-wave mixing in a system of coupled microring resonators (photonic molecule) is developed so that the qubit state is fully controlled by the frequency amplitude of the pump field. It is shown that coupling parameters of the photonic molecule can be optimized to provide generation of nearly pure heralded photons. The heralded qubit state does not depend on the microring resonator mode frequency, thereby allowing to take advantage of frequency multiplexing for improving efficiency of quantum information processing., Comment: Discussion of implementation issues is added

We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder., Comment: Final version accepted for publication in Quantum journal

A $\gamma$-ray-nuclear quantum interface is suggested as a new platform for quantum information processing, motivated by remarkable progresses in $\gamma$-ray quantum optics. The main advantages of a $\gamma$ photon over an optical photon lie in its almost perfect detectability and much tighter, potentially sub-angstrom, focusability. Nuclear ensembles hold important advantages over atomic ensembles in a unique combination of high nuclear density in bulk solids with narrow, lifetime-broadening M\"ossbauer transitions even at room temperature. This may lead to the densest long-lived quantum memories and the smallest size photon processors. Here we propose a technique for $\gamma$ photon quantum memory through a Doppler frequency comb, produced by a set of resonantly absorbing nuclear targets that move with different velocities. It provides a reliable storage, an on-demand generation, and a time sequencing of a single $\gamma$ photon. This scheme presents the first $\gamma$-photon-nuclear-ensemble interface opening a new direction of research in quantum information science.

We present the optimal design for an on-chip single-photon source based on spontaneous four-wave mixing in a system of coupled ring microresonators, which provides frequency uncorrelated joint spectral amplitude of the biphoton field and thereby generation of pure single-photon heralded states. A simple method is proposed for suppressing negative dispersion effects by optimizing the controlled spectroscopic parameters of the system. It shown that the optimal coupling parameters, in combination with the optimal spectral width of the pump pulse, give rise to the highest purity of the heralded photons for a given pump linewidth.

We present optical vector network analysis (OVNA) of an isotopically purified $^{166}$Er$^{3+}$:$^7$LiYF$_4$ crystal. The OVNA method is based on generation and detection of modulated optical sideband by using a radio-frequency vector network analyzer. This technique is widely used in the field of microwave photonics for the characterization of optical responses of optical devices such as filters and high-Q resonators. However, dense solid-state atomic ensembles induce a large phase shift on one of the optical sidebands which results in the appearance of extra features on the measured transmission response. We present a simple theoretical model which accurately describes the observed spectra and helps to reconstruct the absorption profile of a solid-state atomic ensemble as well as corresponding change of the refractive index in the vicinity of atomic resonances., Comment: 4 pages, 5 figures