We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z_2 lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.
We compute the topological entropy of the toric code models in arbitrary dimension at finite temperature. We find that the critical temperatures for the existence of full quantum (classical) topological entropy correspond to the confinement-deconfinement transitions in the corresponding Z_2 gauge theories. This implies that the thermal stability of topological entropy corresponds to the stability of quantum (classical) memory. The implications for the understanding of ergodicity breaking in topological phases are discussed.
We examine two proposals for marginally self-correcting quantum memory, the cubic code by Haah and the welded code by Michnicki. In particular, we prove explicitly that they are absent of topological order above zero temperature, as their Gibbs ensembles can be prepared via a short-depth quantum circuit from classical ensembles. Our proof technique naturally gives rise to the notion of free energy associated with excitations. Further, we develop a framework for an ergodic decomposition of Davies generators in CSS codes which enables formal reduction to simpler classical memory problems. We the…
We propose a scheme to manipulate a topological spin qubit which is realized with cold atoms in a one-dimensional optical lattice. In particular, by introducing a quantum opto-electro-mechanical interface, we are able to first transfer a superconducting qubit state to an atomic qubit state and then to store it into the topological spin qubit. In this way, an efficient topological quantum memory could be constructed for the superconducting qubit. Therefore, we can consolidate the advantages of both the noise resistance of the topological qubits and the scalability of the superconducting qubits …
An explicit lattice realization of a non-Abelian topological memory is presented. The correspondence between logical and physical states is seen directly by use of the stabilizer formalism. The resilience of the encoded states against errors is studied and compared to that of other memories. A set of non-topological operations are proposed to manipulate the encoded states, resulting in universal quantum computation. This work provides insight into the non-local encoding non-Abelian anyons provide at the microscopical level, with an operational characterization of the memories they provide.
Dynamics induced by a change of boundary conditions reveals rate-dependent signatures associated with topological properties in one-dimensional Kitaev chain and SSH model. While the perturbation from a change of the boundary propagates into the bulk, the density of topological edge modes in the case of transforming to open boundary condition reaches steady states. The steady-state density depends on the transformation rate of the boundary and serves as an illustration of quantum memory effects in topological systems. Moreover, while a link is physically broken as the boundary condition changes…
We discuss the existence of stable topological quantum memory at finite temperature. At stake here is the fundamental question of whether it is, in principle, possible to store quantum information for macroscopic times without the intervention from the external world, that is, without error correction. We study the toric code in two dimensions with an additional bosonic field that couples to the defects, in the presence of a generic environment at finite temperature: the toric-boson model. Although the coupling constants for the bare model are not finite in the thermodynamic limit, the model h…
Recently, it has become apparent that the thermal stability of topologically ordered systems at finite temperature, as discussed in condensed matter physics, can be studied by addressing the feasibility of self-correcting quantum memory, as discussed in quantum information science. Here, with this correspondence in mind, we propose a model of quantum codes that may cover a large class of physically realizable quantum memory. The model is supported by a certain class of gapped spin Hamiltonians, called stabilizer Hamiltonians, with translation symmetries and a small number of ground states that…
We propose a simple yet efficient mechanism for passive error correction in topological quantum memories. Our scheme relies on driven-dissipative ancilla systems which couple to local excitations (anyons) and make them "sink" in energy, with no required interaction among ancillae or anyons. Through this process, anyons created by some thermal environment end up trapped in potential "trenches" that they themselves generate, which can be interpreted as a "memory foam" for anyons. This self-trapping mechanism provides an energy barrier for anyon propagation, and removes entropy from the memory by…
We investigate the dynamical interplay between topological state of matter and a non-Markovian dissipation, which gives rise to a new and crucial time scale into the system dynamics due to its quantum memory. We specifically study a one-dimensional polaronic topological superconductor with phonon-dressed $p$-wave pairing, when a fast temperature increase in surrounding phonons induces an open-system dynamics. We show that when the memory depth increases, the Majorana edge dynamics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildup behavior, even …
We discuss energy barriers and their relationship to self-correcting quantum memories. We introduce the solid code, a 3-d version of Kitaev's surface code, and then combine several solid codes using a technique called welding. The resulting code is a $[[O(L^3),1,O(L^{\frac{4}{3}})]]$ stabilizer code with an energy barrier of $O(L^{\frac{2}{3}})$, which is an exponential improvement over the previous highest energy barrier in 3-d. No-go results are avoided by breaking microscopic translation invariance.
The uncertainty principle is one of the most important issues that clarify the distinction between classical and quantum theory. This principle sets a bound on our ability to predict the measurement outcome of two incompatible observables precisely. Uncertainty principle can be formulated via Shannon entropies of the probability distributions of measurement outcome of the two observables. It has shown that the entopic uncertainty bound can be improved by considering an additional particle as the quantum memory $B$ which has correlation with the measured particle $A$. In this work we consider t…
We propose a scheme to dynamically realize a quantum memory based on the toric code. The code is generated from qubit systems with typical two-body interactions (Ising, XY, Heisenberg) using periodic, NMR-like, pulse sequences. It allows one to encode the logical qubits without measurements and to protect them dynamically against the time evolution of the physical qubits. A weakly coupled cavity mode mediates a long-range attractive interaction between the stabilizer operators of the toric code, thereby suppressing the creation of thermal anyons. This significantly increases the lifetime of th…
As a prototype model of topological quantum memory, two-dimensional toric code is genuinely immune to generic local static perturbations, but fragile at finite temperature and also after non-equilibrium time evolution at zero temperature. We show that dynamical localization induced by disorder makes the time evolution a local unitary transformation at all times, which keeps topological order robust after a quantum quench. We verify this conclusion by investigating the Wilson loop expectation value and topological entanglement entropy. Our results suggest that the two dimensional topological qu…
Current approaches for building quantum computing devices focus on two-level quantum systems which nicely mimic the concept of a classical bit, albeit enhanced with additional quantum properties. However, rather than artificially limiting the number of states to two, the use of d-level quantum systems (qudits) could provide advantages for quantum information processing. Among other merits, it has recently been shown that multi-level quantum systems can offer increased stability to external disturbances - a key problem in current technologies. In this study we demonstrate that topological quant…
A local Hamiltonian with Topological Quantum Order (TQO) has a robust ground state degeneracy that makes it an excellent quantum memory candidate. This memory can be corrupted however if part of the state leaves the protected ground state manifold and returns later with a dynamically accrued phase error. Here we analyse how TQO suppresses this process and use this to quantify the degree to which spectral densities in different topological sectors are correlated. We provide numerical verification of our results by modelling an interacting p-wave superconducting wire.
We analyse stability of the four-dimensional Kitaev model - a candidate for scalable quantum memory - in finite temperature within the weak coupling Markovian limit. It is shown that, below a critical temperature, certain topological qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their construction involves polynomial in system's size algorithm which uses as an input the results of measurements performed on all individual spins. We also discuss the drawbacks of such candidate for quantum memory and mention the implications of the stability of q…
Here we investigate the effect lattice geometry has on the lifetime of two-dimensional topological quantum memories. Initially, we introduce various lattice patterns and show how the error-tolerance against bit-flips and phase-flips depends on the structure of the underlying lattice. Subsequently, we investigate the dependence of the lifetime of the quantum memory on the structure of the underlying lattice when it is subject to a finite temperature. Importantly, we provide a simple effective formula for the lifetime of the memory in terms of the average degree of the lattice. Finally, we propo…
Uncertainty principle has a fundamental role in quantum theory. This principle states that it is not possible to measure two incompatible observers simultaneously. The uncertainty principle is expressed logically in terms of standard deviation of the measured observables. In quantum information it has been shown that the uncertainty principle can be expressed by means of Shannon's entropy. Entropic uncertainty relation can be improve by consider an additional particle as the quantum memory. In the presence of quantum memory the entropic uncertainty relation is called quantum memory assisted en…
A general scheme for building a quantum memory by transferring quantum information to an essentially decoherence-free memory transition using quantum control is presented and illustrated by computer simulations.
This thesis addresses whether it is possible to build a robust memory device for quantum information. A three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum informatio…
To realize long-distance quantum communication and quantum network, it is required to have multiplexed quantum memory with many memory cells. Each memory cell needs to be individually addressable and independently accessible. Here we report an experiment that realizes a multiplexed DLCZ-type quantum memory with 225 individually accessible memory cells in a macroscopic atomic ensemble. As a key element for quantum repeaters, we demonstrate that entanglement with flying optical qubits can be stored into any neighboring memory cells and read out after a programmable time with high fidelity. Exper…
Linear-optical systems can implement photonic quantum walks that simulate systems with nontrivial topological properties. Here, such photonic walks are used to jointly entangle polarization and winding number. This joint entanglement allows information processing tasks to be performed with interactive access to a wide variety of topological features. Topological considerations are used to suppress errors, with polarization allowing easy measurement and manipulation of qubits. We provide three examples of this approach: production of two-photon systems with entangled winding number (including t…
We show that universal quantum computation can be performed within the ground state of a topologically ordered quantum system, which is a naturally protected quantum memory. In particular, we show how this can be achieved using brane-net condensates in 3-colexes. The universal set of gates is implemented without selective addressing of physical qubits and, being fully topologically protected, it does not rely on quasiparticle excitations or their braiding.
In this study, the effect of bounded quantum memory in a primitive information protocol has been examined using the quantum Kolmogorov complexity as a measure of information. We employed a toy two-party protocol in which Bob by using a bounded quantum memory and an unbounded classical memory estimates a message that was encoded in qubits by Alice in one of the bases X or Z. Our theorem gave a nontrivial effect of the memory boundedness. In addition, a generalization of the uncertainty principle in the presence of quantum memory has been obtained.
Anderson localization emerges in quantum systems when randomised parameters cause the exponential suppression of motion. Here we consider this phenomenon in topological models and establish its usefulness for protecting topologically encoded quantum information. For concreteness we employ the toric code. It is known that in the absence of a magnetic field this can tolerate a finite initial density of anyonic errors, but in the presence of a field anyonic quantum walks are induced and the tolerable density becomes zero. However, if the disorder inherent in the code is taken into account, we dem…
We consider quantum-memory assisted protocols for discriminating quantum channels. We show that for optimal discrimination of memory channels, memory assisted protocols are needed. This leads to a new notion of distance for channels with memory. For optimal discrimination and estimation of sets of unitary channels memory-assisted protocols are not required.
Using the axiomatic definition of the coherence measure, such as the $l_{1}$ norm and the relative entropy, we study the phenomena of two-qubit system quantum coherence through quantum channels where successive uses of the channels are memory. Different types of noisy channels with memory, such as amplitude damping, phase-damping, and depolarizing channels effect on quantum coherence have been discussed in detail. The results show that, quantum channels with memory can efficiently protect coherence from noisy channels. Particularly, as channels with perfect memory, quantum coherence is unaffec…
In this paper we investigate one dimensional quantum walks with two-step memory, which can be viewed as an extension of quantum walks with one-step memory. We develop a general formula for the amplitudes of the two-step-memory walk with Hadamard coin by using path integral approach, and numerically simulate its process. The simulation shows that the probability distribution of this new walk is different from that of the Hadamard quantum walk with one-step memory, while it presents some similarities with that of the normal Hadamard quantum walk without memory.
Compared to quantum logic gates, quantum memory has received far less attention. Here, we explore the prognosis for a solid-state, scalable quantum dynamic random access memory (Q-DRAM), where the qubits are encoded by the spin orientations of single quantons in exchange-decoupled quantum dots. We address, in particular, various possibilities for implementing refresh cycles.
This paper combines quantum computation with classical neural network theory to produce a quantum computational learning algorithm. Quantum computation uses microscopic quantum level effects to perform computational tasks and has produced results that in some cases are exponentially faster than their classical counterparts. The unique characteristics of quantum theory may also be used to create a quantum associative memory with a capacity exponential in the number of neurons. This paper combines two quantum computational algorithms to produce such a quantum associative memory. The result is an…
Random access memory is an indispensable device for classical information technology. Analog to this, for quantum information technology, it is desirable to have a random access quantum memory with many memory cells and programmable access to each cell. We report an experiment that realizes a random access quantum memory of 105 qubits carried by 210 memory cells in a macroscopic atomic ensemble. We demonstrate storage of optical qubits into these memory cells and their read-out at programmable times by arbitrary orders with fidelities exceeding any classical bound. Experimental realization of …
The storage and retrieval of photonic quantum states, quantum memory, is a key resource for a wide range of quantum applications. Here we investigate the sensitivity of $Λ$-type quantum memory to experimental fluctuations and drift. We use a variance-based approach, focusing on the effects of fluctuations and drift on memory efficiency. We consider shot-to-shot fluctuations of the memory parameters, and separately we consider longer timescale drift of the control field parameters. We find the parameters that a quantum memory is most sensitive to depend on the quantum memory protocol being empl…
Memory effects play a fundamental role in the study of the dynamics of open quantum systems. There exist two conceptually distinct notions of memory discussed for quantum channels in the literature. In quantum information theory quantum channels with memory are characterised by the existence of correlations between successive applications of the channel on a sequence of quantum systems. In open quantum systems theory memory effects arise dynamically during the time evolution of quantum systems, and define non-Markovian dynamics. Here we relate and combine these two different concepts of memory…
Quantum memories are a fundamental of any global-scale quantum Internet, high-performance quantum networking and near-term quantum computers. A main problem of quantum memories is the low retrieval efficiency of the quantum systems from the quantum registers of the quantum memory. Here, we define a novel quantum memory called high-retrieval-efficiency (HRE) quantum memory for near-term quantum devices. An HRE quantum memory unit integrates local unitary operations on its hardware level for the optimization of the readout procedure and utilizes the advanced techniques of quantum machine learnin…
Quantum protocols for bit commitment have been proposed and it is largely accepted that unconditionally secure quantum bit commitment is not possible; however, it can be more secure than classical bit commitment. In despite of its usefulness, quantum bit commitment protocols have not been experimentally implemented. The main reason is the fact that all proposed quantum bit commitment protocols require quantum memory. In this work, we show a quantum bit commitment protocol that does not require quantum memory and can be implemented with present technology.
Quantum memory is a crucial component of a quantum information processor, just like a classical memory is a necessary ingredient of a conventional computer. Moreover, quantum memory of light would serve as a quantum repeater needed for quantum communication networks. Here we propose to realize the quantum memory coupled to magnetic modes of electromagnetic radiation (e.g. those found in cavities and optical fibers) using vortices in Quantum Hall systems. We describe the response to an external magnetic mode as a quench, and find that it sets an oscillation between vortices and antivortices, wi…
In a work by Raz (J. ACM and FOCS 16), it was proved that any algorithm for parity learning on $n$ bits requires either $Ω(n^2)$ bits of classical memory or an exponential number (in~$n$) of random samples. A line of recent works continued that research direction and showed that for a large collection of classical learning tasks, either super-linear classical memory size or super-polynomially many samples are needed. However, these results do not capture all physical computational models, remarkably, quantum computers and the use of quantum memory. It leaves the possibility that a small piece …
Quantum entanglement of particles is regarded as a fundamental character in quantum information, in which quantum state should be given for whole system instead of independently describing single particle. Constructing quantum memory of photonic entanglement is essential for realizing quantum networks, which had been performed previously by many memory protocols. Of which Raman quantum memory gives advantages in broadband and high-speed properties, resulting in huge potential in quantum network and quantum computation. However, Raman quantum memory of photonic polarised entanglement is a chall…
In quantum cryptography the optimal eavesdropping strategy requires that the eavesdropper uses quantum memories in order to optimize her information. What happens if the eavesdropper has no quantum memory? It is shown that the best strategy is actually to adopt the simple intercept/resend strategy.
Within the framework of quantum memory channels we introduce the notion of repeatability of quantum channels. In particular, a quantum channel is called repeatable if there exist a memory device implementing the same channel on each individual input. We show that random unitary channels can be implemented in a repeatable fashion, whereas the nonunital channels cannot.
We present a method to detect quantum memory in a non-Markovian process. We call a process Markovian when the environment does not provide a memory that retains correlations across different system-environment interactions. We define two types of non-Markovian processes, depending on the required memory being classical or quantum. We formalise this distinction using the process matrix formalism, through which a process is represented as a multipartite state. Within this formalism, a test for entanglement in a state can be mapped to a test for quantum memory in the corresponding process. This a…
We propose a scheme of a universal block of broadband quantum memory consisting of three ring microresonators forming a controllable frequency comb and interacting with each other and with a common waveguide. We find the optimal parameters of the microresonators showing the possibility of highly efficient storage of light fields on this memory block and we demonstrate the procedure for gluing several memory blocks for increasing spectral range of the composite quantum memory while maintaining high efficiency.
Photonic quantum memory, such as an atomic frequency comb (AFC), is essential to make photonic quantum computation and long distance quantum communication scalable and feasible. In standard AFC the frequency of different atoms must be stable relative to each other which presents difficulties in realizing the quantum memory. Here we propose a quantum memory using an intra-atomic frequency comb which does not require frequency stabilization. We show that the transitions between two degenerate energy levels of a single atom can be used to construct the frequency comb. The spacing between the teet…
A "bucket brigade" architecture for a quantum random memory of $N=2^n$ memory cells needs $n(n+5)/2$ times of quantum manipulation on control circuit nodes per memory call. Here we propose a scheme, in which only average $n/2$ times manipulation is required to accomplish a memory call. This scheme may significantly decrease the time spent on a memory call and the average overall error rate per memory call. A physical implementation scheme for storing an arbitrary state in a selected memory cell followed by reading it out is discussed.
The uncertainty principle restricts potential information one gains about physical properties of the measured particle. However, if the particle is prepared in entanglement with a quantum memory, the corresponding entropic uncertainty relation will vary. Based on the knowledge of correlations between the measured particle and quantum memory, we have investigated the entropic uncertainty relations for two and multiple measurements, and generalized the lower bounds on the sum of Shannon entropies without quantum side information to those that allow quantum memory. In particular, we have obtained…
We study the effect of quantum memory in non-inertial frames under the influence of amplitude damping, depolarizing, phase flip and bit-phase flip channels. It is shown that the entanglement of initial state is heavily influenced by quantum correlations. It is seen that quantum memory compensates the loss of entanglement generated caused by the Unruh effect. It is interesting to note that the sudden death of entanglement disappears for any acceleration for higher values of memory. Therefore, it is possible to avoid ESD in non-inertial frames due to the presence of quantum memory. Furthermore, …
