A formulation for performing quantum computing in a projected subspace is presented, based on the subdynamical kinetic equation (SKE) for an open quantum system. The eigenvectors of the kinetic equation are shown to remain invariant before and after interaction with the environment. However, the eigenvalues in the projected subspace exhibit a type of phase shift to the evolutionary states. This phase shift does not destroy the decoherence-free (DF) property of the subspace because the associated fidelity is 1. This permits a universal formalism to be presented - the eigenprojectors of the free part of the Hamiltonian for the system and bath may be used to construct a DF projected subspace based on the SKE. To eliminate possible phase or unitary errors induced by the change in the eigenvalues, a cancellation technique is proposed, using the adjustment of the coupling time, and applied to a two qubit computing system. A general criteria for constructing a DF projected subspace from the SKE is discussed. Finally, a proposal for using triangulation to realize a decoherence-free subsystem based on SKE is presented. The concrete formulation for a two-qubit model is given exactly. Our approach is novel and general, and appears applicable to any type of decoherence. Key Words: Quantum Computing, Decoherence, Subspace, Open System PACS number: 03.67.Lx,33.25.+k,.76.60.-k
Using the subdynamical kinetic equation for an open quantum system, a formulation is presented for performing decoherence-free (DF) quantum computing in Rigged Liouville Space (RLS). Three types of interactions were considered, and in each case, stationary and evolutionary states were evaluated for DF behavior in both the total space and the projected subspace. Projected subspaces were found using the subdynamics kinetic equation. It was shown that although the total space may be decoherent, the subspace can be DF. In the projected subspace, the evolution of the density operator may be time as…
We discuss a general physical mechanism for arbitrary control of the quantum states of multiple qubits in the symmetric Dicke subspace. The qubit-qubit coupling leads to unequal energy spacing in the symmetric Dicke subspace. This allows one to manipulate a prechosen transition with an external driving source, with other transitions remaining off-resonant. Any entangled state in the symmetric Dicke subspace can be created from the initial ground state by tuning the driving source. We illustrate the idea in cavity QED, but it should be applicable to other related systems.
Non-adiabatic holonomic quantum gate in decoherence-free subspaces is of greatly practical importance due to its built-in fault tolerance, coherence stabilization virtues, and short run-time. Here we propose some compact schemes to implement two- and three-qubit controlled unitary quantum gates and Fredkin gate. For the controlled unitary quantum gates, the unitary operator acting on the target qubit is an arbitrary single-qubit gate operation. The controlled quantum gates can be directly implemented using non-adiabatic holonomy in decoherence-free subspaces and the required resource for the d…
The basic operating element of standard quantum computation is the qubit, an isolated two-level system that can be accurately controlled, initialized and measured. However, the majority of proposed physical architectures for quantum computation are built from systems that contain much more complicated Hilbert space structures. Hence, defining a qubit requires the identification of an appropriate controllable two-dimensional sub-system. This prompts the obvious question of how well a qubit, thus defined, is confined to this subspace, and whether we can experimentally quantify the potential leak…
We propose a geometric phase gate in a decoherence-free subspace with trapped ions. The quantum information is encoded in the Zeeman sublevels of the ground-state and two physical qubits to make up one logical qubit with ultra long coherence time. Single- and two-qubit operations together with the transport and splitting of linear ion crystals allow for a robust and decoherence-free scalable quantum processor. For the ease of the phase gate realization we employ one Raman laser field on four ions simultaneously, i.e. no tight focus for addressing. The decoherence-free subspace is left neither …
We study the local distinguishability of general multi-qubit states and show that local projective measurements and classical communication are as powerful as the most general local measurements and classical communication. Remarkably, this indicates that the local distinguishability of multi-qubit states can be decided efficiently. Another useful consequence is that a set of orthogonal $n$-qubit states is locally distinguishable only if the summation of their orthogonal Schmidt numbers is less than the total dimension $2^n$. When $n=2$ such a condition is also sufficient. Employing these resu…
We propose a many-qubit network with cavity QED by encoding qubits in decoherence-free subspace, based on which we can implement many-logic-qubit conditional gates by means of cavity assisted interaction with single-photon pulses. Our scheme could not only resist collective dephasing errors, but also much reduce the implementational steps compared to conventional methods doing the same job, and we can also complete communications between two arbitrary nodes. We show the details by implementing a three-logic-qubit Toffoli gate, and explore the experimental feasibility and challenge based on cur…
The whole Hilbert state space of an n-qubit spin system can be divided into (n+1) state subspaces according to the angular momentum theory of quantum mechanics. Here it is shown that any unknown state in such a state subspace, whose dimensional size is proportional to either a polynomial or exponential function of the qubit number n, can be transferred efficiently into a larger subspace with a dimensional size generally proportional to an exponential function of the qubit number by the multiple-quantum unitary transformation with a subspace-selective multiple-quantum unitary operator. The effi…
We investigate how to carry out universal quantum computation deterministically with free electrons in decoherence-free subspace by using polarizing beam splitters, charge detectors, and single-spin rotations. Quantum information in our case is encoded in spin degrees of freedom of the electron-pairs which construct a decoherence-free subspace. We design building blocks for two noncommutable single-logic-qubit gates and a logic controlled phase gate, based on which a universal and scalable quantum information processing robust to dephasing is available in a deterministic way.
We describe an alternative approach to quantum computation that is ideally suited for today's sub-threshold-fidelity qubits, and which can be applied to a family of hardware models that includes superconducting qubits with tunable coupling. In this approach, the computation on an n-qubit processor is carried out in the n-dimensional single-excitation subspace (SES) of the full 2^n-dimensional Hilbert space. Because any real Hamiltonian can be directly generated in the SES [E. J. Pritchett et al., arXiv:1008.0701], high-dimensional unitary operations can be carried out in a single step, bypassi…
Coherence in an open quantum system is degraded through its interaction with a bath. This decoherence can be avoided by restricting the dynamics of the system to special decoherence-free subspaces. These subspaces are usually constructed under the assumption of spatially symmetric system-bath coupling. Here we show that decoherence-free subspaces may appear without spatial symmetry. Instead, we consider a model of system-bath interactions in which to first order only multiple-qubit coupling to the bath is present, with single-qubit system-bath coupling absent. We derive necessary and sufficien…
The rotation of subspaces by a chosen angle is a fundamental quantum computing operation, with applications in error correction and quantum algorithms such as the Quantum Approximate Optimization Algorithm, the Variational Quantum Eigensolver and the quantum singular value transformation. Such rotations are usually implemented at the hardware level via multiple-controlled-phase gates, which lead to large circuit depth when decomposed into one- and two-qubit gates. Here, we propose a fast, high-fidelity way to implement such operations via topological quantum walks, where a sequence of single-q…
When the environmental disturbace to a quantum system has a wavelength much larger than the system size, all qubits localized within a small area are under action of the same error operators. Noiseless subsystem and decoherence free subspace are known to correct such collective errors. We construct simple quantum circuits, which implement these collective error correction codes, for a small number $n$ of physical qubits. A single logical qubit is encoded with $n=3$ and $n=4$, while two logical qubits are encoded with $n=5$. The recursive relations among the subspaces employed in noiseless subs…
We investigate the groups generated by the sets of $CP$, $CNOT$ and $SWAP^α$ (power-of-SWAP) quantum gate operations acting on $n$ qubits. Isomorphisms to standard groups are found, and using techniques from representation theory, we are able to determine the invariant subspaces of the $n-$qubit Hilbert space under the action of each group. For the $CP$ operation, we find isomorphism to the direct product of $n(n-1)/2$ cyclic groups of order $2$, and determine $2^n$ $1$-dimensional invariant subspaces corresponding to the computational state-vectors. For the $CNOT$ operation, we find isomorphi…
Different from the previous works on generating entangled states, this work is focused on how to transfer the prepared entangled states onto memory qubits for protecting them against decoherence. We here consider a physical system consisting of $n$ operation qubits and $2n$ memory qubits placed in a cavity or coupled to a resonator. A method is presented for transferring $n$-qubit Greenberger-Horne-Zeilinger (GHZ) entangled states from the operation qubits (i.e., information processing cells) onto the memory qubits (i.e., information memory elements with long decoherence time). The transferred…
A scheme to implement a quantum computer subjected to decoherence and governed by an untunable qubit-qubit interaction is presented. By concatenating dynamical decoupling through bang-bang (BB) pulse with decoherence-free subspaces (DFSs) encoding, we protect the quantum computer from environment-induced decoherence that results in quantum information dissipating into the environment. For the inherent qubit-qubit interaction that is untunable in the quantum system, BB control plus DFSs encoding will eliminate its undesired effect which spoils quantum information in qubits. We show how this qua…
We present a new robust decoupling scheme suitable for levels with either half integer or integer angular momentum states. Through continuous dynamical decoupling techniques, we create a protected qubit subspace, utilizing a multi-state qubit construction. Remarkably, the multi-state system can also be comprised of multiple sub-states within a single level. Our scheme can be realized with state-of-the-art experimental setups and thus has immediate applications for quantum information science. While the scheme is general and relevant for a multitude of solid state and atomic systems, we analyze…
We present a formalism for encoding the logical basis of a qubit into subspaces of multiple physical levels. The need for this multilevel encoding arises naturally in situations where the speed of quantum operations exceeds the limits imposed by the addressability of individual energy levels of the qubit physical system. A basic feature of the multilevel encoding formalism is the logical equivalence of different physical states and correspondingly, of different physical transformations. This logical equivalence is a source of a significant flexibility in designing logical operations, while the…
We propose a criterion for the detection of genuine entanglement of pure multiqubit states. To this aim, we define an operator called the losing one qubit operator, which is different from the reduced density operator. The states obtained from a multiqubit state by applying the losing one qubit operator are referred to as its projected states. We show that all of the projected states of a pure product n-qubit state are pure product states provided that it cannot be written as a product of a single qubit state and a genuinely entangled (n-1)-qubit state. We also show that a pure n-qubit state i…
Protecting quantum states from the decohering effects of the environment is of great importance for the development of quantum computation devices and quantum simulators. Here, we introduce a continuous dynamical decoupling protocol that enables us to protect the entangling gate operation between two qubits from the environmental noise. We present a simple model that involves two qubits which interact with each other with a strength that depends on their mutual distance and generates the entanglement among them, as well as in contact with an environment. The nature of the environment, that is,…
Decoherence-free subspaces (DFSs) shield quantum information from errors induced by the interaction with an uncontrollable environment. Here we study a model of correlated errors forming an Abelian subgroup (stabilizer) of the Pauli group (the group of tensor products of Pauli matrices). Unlike previous studies of DFSs, this type of errors does not involve any spatial symmetry assumptions on the system-environment interaction. We solve the problem of universal, fault-tolerant quantum computation on the associated class of DFSs.
Quantum state transfer and teleportation, with qubits encoded in internal states of the atoms in cavities, among spatially separated nodes of a quantum network in decoherence-free subspace are proposed, based on a cavity-assisted interaction by single-photon pulses. We show in details the implementation of a logic-qubit Hadamard gate and a two-logic-qubit conditional gate, and discuss the experimental feasibility of our scheme.
We propose a scheme to implement quantum computation in decoherence-free subspace with superconducting devices inside a cavity by unconventional geometric manipulation. Universal single-qubit gates in encoded qubit can be achieved with cavity assisted interaction. A measurement-based two-qubit Controlled-Not gate is produced with parity measurements assisted by an auxiliary superconducting device and followed by prescribed single-qubit gates. The measurement of currents on two parallel devices can realize a projective measurement, which is equivalent to the parity measurement on the involved d…
We experimentally demonstrate the efficacy of a three-layer nested Uhrig dynamical decoupling (NUDD) sequence to preserve arbitrary quantum states in a two-dimensional subspace of the four-dimensional two-qubit Hilbert space, on an NMR quantum information processor. The effect of the state preservation is studied first on four known states, including two product states and two maximally entangled Bell states. Next, to evaluate the preservation capacity of the NUDD scheme, we apply it to eight randomly generated states in the subspace. Although, the preservation of different states varies, the …
We investigate the problem of teleporting an unknown qubit state to a recipient via a channel of $2Ł$ qubits. In this procedure a protocol is employed whereby $Ł$ Bell state measurements are made and information based on these measurements is sent via a classical channel to the recipient. Upon receiving this information the recipient determines a local gate which is used to recover the original state. We find that the $2^{2Ł}$-dimensional Hilbert space of states available for the channel admits a decomposition into four subspaces. Every state within a given subspace is a perfect channel, and e…
A permutation-invariant code on m qubits is a subspace of the symmetric subspace of the m qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading order spontaneous decay errors. To prove the result, we use elementary number theory with prior theory on permutation invariant codes and quantum error correction.
We introduce a discrete Q-function of N qubit system projected into the space of symmetric measurements as a tool for analyzing general properties of quantum systems in the macroscopic limit. For known states the projected Q-function helps to visualize the results of collective measurements, and for unknown states it can be approximately reconstructed by measuring lowest moments of the of collective variables.
We present a new and simplified two-qubit randomized benchmarking procedure that operates only in the symmetric subspace of a pair of qubits and is well suited for benchmarking trapped-ion systems. By performing benchmarking only in the symmetric subspace, we drastically reduce the experimental complexity, number of gates required, and run time. The protocol is demonstrated on trapped ions using collective single-qubit rotations and the Molmer-Sorenson (MS) interaction to estimate an entangling gate error of $2(1) \times 10^{-3}$. We analyze the expected errors in the MS gate and find that pop…
Population leakage outside the qubit subspace presents a particularly harmful source of error that cannot be handled by standard error correction methods. Using a trapped $^{171}$Yb$+$ ion, we demonstrate an optical pumping scheme to suppress leakage errors in atomic hyperfine qubits. The selection rules and narrow linewidth of a quadrupole transition are used to selectively pump population out of leakage states and back into the qubit subspace. Each pumping cycle reduces the leakage population by a factor of $\sim3$, allowing for an exponential suppression in the number of cycles. We use inte…
Repeated observations inhibit the coherent evolution of quantum states through the quantum Zeno effect. In multi-qubit systems this effect provides new opportunities to control complex quantum states. Here, we experimentally demonstrate that repeatedly projecting joint observables of multiple spins creates coherent quantum Zeno subspaces and simultaneously suppresses dephasing caused by the environment. We encode up to two logical qubits in these subspaces and show that the enhancement of the dephasing time with increasing number of projections follows a scaling law that is independent of the …
There is currently a tremendous interest in developing practical applications of NISQ processors without the overhead required by full error correction. Quantum information processing is especially challenging within the gate model, as algorithms quickly lose fidelity as the problem size and circuit depth grow. This has lead to a number of non-gate-model approaches such as analog quantum simulation and quantum annealing. These approaches come with specific hardware requirements that are typically different than that of a universal gate-based quantum computer. We have previously proposed a non-…
A generalized universal quantum cloning machine is proposed which allows the input to be arbitrary states in symmetric subspace. And it reduces to the universal quantum cloning machine (UQCM) if the input are identical pure states. The generalized cloner is optimal in the sense we compare the input and output reduced density operators at a single qubit. The result for qubits is extended to arbitrary-dimensional states.
A fundamental requirement of quantum information processing is the protection from the adverse effects of decoherence and noise. Decoherence-free subspaces and geometric processing are important steps of quantum information protection. Here, we provide a new experimentally feasible scheme to combine decoherence-free subspaces with nonadiabatic geometric manipulations to attain a universal quantum computation. The proposed scheme is different from previous proposals and is based on the typical XY interaction coupling, which can be set up in various nano-engineered systems and therefore open up …
We propose a scheme to implement geometric entangling gates for two logical qubits in a coupled cavity system in decoherence-free subspaces. Each logical qubit is encoded with two atoms trapped in a single cavity and the geometric entangling gates are achieved by cavity coupling and controlling the external classical laser fields. Based on the coupled cavity system, the scheme allows the scalability for quantum computing and relaxes the requirement for individually addressing atoms.
The quantum dynamics of the Josephson junction system in the computational subspace is investigated. A scheme for the controlled not operation is given for two capasitively coupled SQUIDs. In this system, there is no systematic error for the two qubit operation. For the inductively coupled SQUIDs, the effective Hamiltonian causes systematic errors in the computational subspace for the two qubit operation. Using the purterbation theory, we construct a more precise effective Hamiltonian. This new effective Hamiltonian reduces the systematic error to the level much lower than the threshold of the…
Nonadiabatic holonomic quantum computation in decoherence-free subspaces has attracted increasing attention recently, as it allows for high-speed implementation and combines both the robustness of holonomic gates and the coherence stabilization of decoherence-free subspaces. Since the first protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces, a number of schemes for its physical implementation have been put forward. However, all previous schemes require two noncommuting gates to realize an arbitrary one-qubit gate, which doubles the exposure time of gates to er…
Evolution of quantum states of array of quantum dots is analyzed by means of numerical solution of the von Neumann equation. For two qubit system with dipole-dipole interaction and common phonon bath the evolution of the symmetric state $\frac{\uparrow\downarrow+\downarrow\uparrow}{\sqrt{2}}$ leads to the mixture of the triplet states, leaving the singlet decoupled. For three qubit system ($D_{1/2}^{\otimes3}=D_{3/2}+2D_{1/2}$) with common phonon bath we observed similar effects within the quartet state $D_{3/2}$ if all qubits were symmetrically connected.
We study a quantum repeater which is based on decoherence free quantum gates recently proposed by Klein et al. [Phys. Rev. A 73, 012332 (2006)]. A number of operations on the decoherence free subspace in this scheme makes use of an ancilla qubit, which undergoes dephasing and thus introduces decoherence to the system. We examine how this decoherence affects entanglement swapping and purification as well as the performance of a quantum repeater. We compare the decoherence free quantum repeater with a quantum repeater based on qubits that are subject to decoherence and show that it outperforms t…
Learning with Active Quantum Subspaces: Scalable Hybrid Advantage without Full Quantum Data-Encoding
Yonsei University（延世大学）；Korea Institute of Science and Technology Information（韩国科学技术信息研究所）；Electronics and Telecommunications Research Institute（电子电信研究院）
Quantum computing promises significant speed-up for certain types of computational problems. However, robust implementations of semiconducting qubits must overcome the effects of charge noise that currently limit coherence during gate operations. Here we describe a scheme for protecting solid-state qubits from uniform electric field fluctuations by generalizing the concept of a decoherence-free subspace for spins, and we propose a specific physical implementation: a quadrupole charge qubit formed in a triple quantum dot. The unique design of the quadrupole qubit enables a particularly simple p…
We propose and analyse a robust quantum state transfer protocol by the use of a combination of coherent quantum coupling and decoherence-free subspaces in a coupled quantum spin chain. Under decoherence, an arbitrary unknown quantum state embedded in a decoherence-free subspace can be perfectly transferred through a noisy channel being weakly coupled to two end registers. The method protects quantum information from both the channel noise and the environmental decoherence. A special case of utilizing two physical qubits to encode one logical qubit is considered as an example and the robustness…
There is a natural orthogonal basis of the 6-qubit decoherence-free (DF) space robust against collective noise. Interestingly, most of the basis states can be obtained from one another just permuting qubits. This property: (a) is useful for encoding qubits in DF subspaces, (b) allows the implementation of the Bennett-Brassard 1984 (BB84) protocol in DF subspaces just permuting qubits, which completes a the method for quantum key distribution using DF states proposed by Boileau et al. [Phys. Rev. Lett. 92, 017901 (2004)], and (c) points out that there is only one 6-qubit DF state which is essen…
Quantum computation that combines the coherence stabilization virtues of decoherence-free subspaces and the fault tolerance of geometric holonomic control is of great practical importance. Some schemes of adiabatic holonomic quantum computation in decoherence-free subspaces have been proposed in the past few years. However, non-adiabatic holonomic quantum computation in decoherence-free subspaces, which avoids long run-time requirement but with all the robust advantages, remains an open problem. Here, we demonstrate how to realize non-adiabatic holonomic quantum computation in decoherence-free…
The longitudinal coupling of a system to the bath usually induces the pure dephasing of the system. In this paper, we study the collective dephasing induced dissipation and decoherence in a coupled-qubit system with a common bath. It is shown that, compared with the case of the same system with independent baths, the interference between the dephasing processes of different qubits induced by the common bath significantly changes the dissipation of the system. For the system of two coupled qubits, the interference leads to a faster decoherence in the non-single-excitation subspaces and a slower…
A general scheme to perform universal quantum computation within decoherence-free subspaces (DFSs) of a system's Hilbert space is presented. This scheme leads to the first fault-tolerant realization of universal quantum computation on DFSs with the properties that (i) only one- and two-qubit interactions are required, and (ii) the system remains within the DFS throughout the entire implementation of a quantum gate. We show explicitly how to perform universal computation on clusters of the four-qubit DFS encoding one logical qubit each under "collective decoherence" (qubit-permutation-invariant…
It is possible to prepare a composite qubit-environment system so that its time evolution will guarantee the conservation of a preselected qubit's observable. In general, this observable is not associated with a symmetry. The latter may not even be present in the subsystem. The initial states which lead to such a quantity conserved dynamics form a subspace of the qubit-environment space of states. General construction of this subspace is presented and illustrated by two examples. The first one is the exactly soluble Jaynes-Cummings model and the second is the multi-photon Rabi model.
We study the energy level crossings of the states and thermal fidelity for a two-qubit system in the presence of a transverse and inhomogeneous magnetic field. It is shown clearly the effects of the anisotropic factor of the magnetic field through the contour figures of energy level crossing in two subspaces, the isotropy subspace and anisotropy subspace. We calculate the quantum fidelity between the ground state and the state of the system at temperature $T$, and the results show the strong effect of the anisotropic factor again. In addition, by making use of the transition of Yangian generat…
Genuine entanglement is the strongest form of multipartite entanglement. Genuinely entangled pure states contain entanglement in every bipartition and as such can be regarded as a valuable resource in the protocols of quantum information processing. A recent direction of research is the construction of genuinely entangled subspaces -- the class of subspaces consisting entirely of genuinely multipartite entangled pure states. In this paper we present several methods of construction of such subspaces including those of maximal possible dimension. The approach is based on the correspondence betwe…
Geometric phases have been used in NMR, to implement controlled phase shift gates for quantum information processing, only in weakly coupled systems in which the individual spins can be identified as qubits. In this work, we implement controlled phase shift gates in strongly coupled systems, by using non-adiabatic geometric phases, obtained by evolving the magnetization of fictitious spin-1/2 subspaces, over a closed loop on the Bloch sphere. The dynamical phase accumulated during the evolution of the subspaces, is refocused by a spin echo pulse sequence and by setting the delay of transition …
A parity measurement on two qubits, each consisting of a single atom in a cavity, can be realized by measuring the phase shift of a probe beam, which interacts sequentially with the two qubits, but imperfections lead to decoherence within the subspaces of a given parity. We demonstrate that a different setup, where the probe light interacts repeatedly with the qubits, can reduce the rate of decoherence within the odd or the even parity subspace significantly. We consider both the case of a resonant and the case of a nonresonant light-atom interaction and find that the performance is comparable…
