Disentangled Quantum Operation on Majorana Qubits in Qubit Chains

Abstract

We consider quantum logical gates on Majorana qubits implemented in chain structures of ordinary qubits, spins, or pseudospins. We demonstrate that one can implement a two-qubit operation via local manipulations, using an extra coupler spin in a T-junction geometry, so that this coupler spin remains disentangled from the qubit. Furthermore, we identify a set of symmetry operations, which not only allow us to determine the resulting two-qubit gate, but also to demonstrate robustness of the resulting gate to inaccuracies in the manipulations, known for topological quantum computation.

INTRODUCTION

Majorana degrees of freedom in condensed matter are studied extensively [1, 2]. Due to their unusual physical properties, they can provide the base for topological quantum computation [3–5]. They may arise at edges of one-dimensional systems in topological phases due to the bulk-boundary correspondence. To implement not only topologically protected qubits for storing quantum information but also topologically protected quantum logical gates, a braiding operation for the Majorana fermions [6, 7] was suggested [3]. Implementation of a braiding operation with one-dimensional wires was proposed in [8, 9] in a setup of a 2D wire network, where a basic ingredient used for braiding was a T-junction.

The Jordan–Wigner transformation [10] connects fermionic and spin degrees of freedom in 1D geometry. It can be generalized to networks of 1D wires, cf. [11–13]. For this reason, observation of fermionic phenomena in spin/qubit networks attracted attention of researchers. Recently, physical simulation of braiding in networks of spin or qubit chains was proposed, with discussion of experimental implementation in circuits of Josephson qubits [12]. This research direction is of interest because of the progress with experiments on quantum bits and due to the growing interest to quantum simulations. In particular, it allows for physical simulation of fermionic degrees of freedom. Let us remark that, although the braiding operation in the fermionic picture is well understood, its simulation in spin/qubit systems requires a description in the spin language. It is useful not only for the design and interpretation of the prospective experiments. More importantly, it is needed for understanding and analysis of the role of various imperfections, including the influence of relevant noise sources, highly nonlocal in the fermionic language, as well as of imperfections in the control pulses. Experimental groups investigate realizations of this and similar topological models [1214].

In the spin language, a qubit is represented by a ferromagnetic interval. Adiabatic manipulations with local fields in the vicinity of this interval may change the state of the qubit. This implements a single-qubit quantum logical gate, which is an analog of the braiding operation for two MZM’s in the fermionic language [8, 12]. Further, for two qubit intervals, an inter-qubit braiding operation was suggested [13], and its result was found for simple initial basis states and for a simplified manipulation procedure. Some analysis of the role of these simplifications was presented in [15]. On the other hand, even for the simplified procedure, it was found that the resulting two-qubit gate entangles the states of the qubits with an auxiliary coupler spin, so that the initial state of the spin and its imperfect dynamics are relevant for the resulting two-qubit gate.

Here we extend earlier results and demonstrate that one can implement a two-qubit quantum logical gate using such methods that the qubits do not get entangled with the coupler spin at the junction, similarly to single-qubit gates. Furthermore, we find a symmetry of the problem, which allows us, on one hand, to find the result of the complete two-qubit gate, not only for the basis initial states but also for their superpositions, and on the other hand, to show robustness of the resulting two-qubit gate towards variations in the manipulation procedure. This implies a high degree of flexibility during the manipulations with a robust resulting gate, exactly in the spirit of topological quantum computation.

TWO-QUBIT GATE VIA A T-JUNCTION

In order to describe our findings, we briefly describe the considered system and the implementation of a two-qubit logical operation. We consider a system of three Ising chains, connected at a junction in the form of a triangle, see Fig. 1. Each site carries a qubit (or spin/pseudo-spin) degree of freedom.

Fig. 1.
figure1

A Δ-junction, connecting three Ising spin chains. The couplings between the spins σα(l) at the chain ends are controlled by the coupler spin S. The Hamiltonian, and hence evolution, of the system is controlled via local pulses, which allows for shifting of (pseudo-)topological intervals. The Δ-junction reduces to a T-junction when one coupling at the triangle vanishes.

The Hamiltonian of the system is given by

$$H = \sum\limits_{\alpha = 1}^3 {{{H}_{\alpha }} + H_{{{\text{jct}}}}^{S},} $$
((1))

where the Hamiltonians of the three chains are

$${{H}_{\alpha }} = - \sum\limits_{i \geqslant 1}^{} {{{h}_{\alpha }}(i)\sigma _{\alpha }^{x}(i)} - J\sum\limits_{i \geqslant 1}^{} {\sigma _{\alpha }^{z}(i)\sigma _{\alpha }^{z}(i + 1),} $$
((2))

while the coupling of the chains at the junction is described by

$$\begin{gathered} H_{{{\text{jct}}}}^{S} = - \frac{1}{2}{{J}_{{12}}}{{S}^{3}}\sigma _{1}^{z}(1)\sigma _{2}^{z}(1) \\ - \frac{1}{2}{{J}_{{23}}}{{S}^{1}}\sigma _{2}^{z}(1)\sigma _{3}^{z}(1) - \frac{1}{2}{{J}_{{13}}}{{S}^{2}}\sigma _{1}^{z}(1)\sigma _{3}^{z}(1). \\ \end{gathered} $$
((3))

In other words, the system contains three Ising spin chains with local transverse magnetic fields hα(i). These fields can be controlled externally. While the sign of the coupling J has no effect on the physics, we assume that J > 0. In each of the Ising chains any region subject to a strong transverse field, |h| ≫ J, is in a trivial paramagnetic phase, whereas the regions in a zero or weak field, |h| ≪ J, are ferromagnetic. We enumerate sites in each chain starting from i = 1. The edge sites, i = 1, in the chains are coupled pairwise via the term (3), and these three coupling terms are controlled by three different components of an extra “coupler” spin S (relevance and implementation of such couplings are discussed in [12, 16–18] for various qubit realizations). This Hamiltonian describes a so called Δ-junction [19] with a symmetric structure of the coupling between the chains. It reduces to a T-junction if one of three couplings in the Δ triangle vanishes.

Similarly to the Jordan-Wigner transformation between spin and fermionic degrees of freedom in a one-dimensional chain [10], a generalized Jordan–Wigner transformation maps this system to a system of free fermions (1) in the same T-junction geometry [11, 12, 20]. Thus one can consider analogs of fermionic phenomena in this spin system. In particular, one can find by this method analogs of Majorana fermionic qubits, based on Majorana zero modes (MZM’s), and the braiding operation with these degrees of freedom, which effects quantum logical gates for such qubits. For instance, a qubit is realized as a low-field region (h ≪ J) between high-field regions (hJ), with two MZM’s at its ends. Using control over the local fields, one can implement the braiding operation for the MZM’s. Indeed, tuning the transverse fields between the low and high values, one can extend or shrink any qubit (low-field) region at either end.

Thus one can shift the ferromagnetic (pseudotopological, or qubit) regions. In particular, by moving a single qubit through a T-junction, see Fig. 2a, one can achieve a single-qubit operation via the braiding of the effective Majorana zero modes at its ends [12].

Fig. 2.
figure2

(Color online) Braiding operations with ferromagnetic qubits. (a) Single-qubit braiding, (b) Two-qubit braiding. During the manipulations the topological regions (thick lines) are shifted between the chains as shown. Arrows indicate transitions between various stages of the operations.

Similarly, by shifting two qubit regions, see Fig. 2b, one can implement a two-qubit operation [13, 15]. One can straightforwardly find the result of this operation for a simple initial state (both qubits as well as the coupler spin in the spin-up state) and a simplified (although not easily implemented) manipulation procedure [13, 15]. While it can easily be generalized to other initial basis states, this leaves certain ambiguity in the resulting two-qubit gate. Indeed, for each initial basis state, the final state after adiabatic manipulations is found only up to a phase factor. Furthermore, the use of the simplified procedure raises questions about the robustness of the resulting gate. We discuss these problems below.

First, let us briefly describe the manipulations needed for the two-qubit operation. The initial state before the operation is indicated in Fig. 2b, with one qubit in the left chain (chain 1) and the other qubit in the right chain (chain 3). During the first stage of the operation, the right end of the first qubit region is moved from chain 1 down to chain 2; the Sz component of the coupler spin is conserved, and the operation is easily described in its eigenbasis. During the second stage, the other qubit in chain 3 is moved towards the junction. In the simplified procedure, this is performed as J13 = J23 = 0, which allows following the manipulation of the spin σ3(1) easily; and only afterwards these two junction couplings are turned on. During the next stage, the transverse field for the spin σ1(1) is increased to freeze this spin in the x direction and the right end of the left qubit is moved back to chain 1; during this stage, the coupler spin adjusts adiabatically to the x direction. Finally, the left end of the right qubit is moved back to chain 3 (as above, this stage is easy to describe in the eigenbasis of Sx).

By following the evolution step by step, one finds the resulting state. Consider the basis states of the first qubit interval with all spins up (a = +1) or down (a = –1) and the same for the second qubit (b = ±1) and the coupler spin (SSz = ±1). If the initial state of the qubits and the coupler was a = +1, b = +1, and S = +1, the final state is found to be |a = +1, b = +1, +\(\hat {x}\)〉, up to a phase factor. In this state the coupler spin is adjusted to the direction +x.

In spite of the ambiguous relative phase for different initial basis states, can one find the complete two-qubit operation? What happens if the manipulation procedure is not followed exactly but only approximately? These questions are discussed below.

SYMMETRIES OF THE PROBLEM AND THE COMPLETE TWO-QUBIT GATE

In this section we demonstrate that the problem of the T-junction is invariant under three symmetry operations, described below. On one hand, this allows us to find the result of the two-qubit braiding operation for all basis states and their superpositions as initial states, based only on its result for one simple initial state discussed above. However, it also implies a certain robustness of the system in the sense discussed below.

These symmetry operations, which conserve the Hamiltonian, are the following: one operation,

$${{R}_{2}} = R_{y}^{S}(\pi ) \cdot \prod\limits_{i \geqslant 1}^{} {R_{x}^{{\sigma _{i}^{2}}}(\pi ),} $$
((4))

flips (z-components of) all the spins in chain 2 as well as the x- and z-components of the coupler spin. Another operation,

$${{R}_{1}} = R_{x}^{S}(\pi ) \cdot \prod\limits_{i \geqslant 1}^{} {R_{x}^{{\sigma _{i}^{1}}}(\pi ),} $$
((5))

flips all the spins in chain 1 as well as the y- and z-components of the coupler spin. Finally, the third operation,

$${{R}_{3}} = R_{z}^{S}(\pi ) \cdot \prod\limits_{i \geqslant 1}^{} {R_{x}^{{\sigma _{i}^{3}}}(\pi ),} $$
((6))

flips all the spins in chain 3 as well as the x- and y-components of the coupler spin.

In summary, each operation flips (z-components of) all the spins in one of the three chains (indicated by the subscript in the notation for the operation) as well as those two components of the coupler spin which control the two couplings adjacent to this chain in the triangle at the junction (these two components of S are indicated in Fig. 1 next to this chain). From this short description, one indeed observes that these operations conserve the Hamiltonian of the T-junction.

These symmetries conserve the Hamiltonian even with some degree of disorder, for instance, when the transverse fields hi are non-uniform. Since they commute with the Hamiltonian, they relate evolutions with initial conditions linked by these symmetries. Indeed, if R is one of these symmetry operations or a combination thereof, then RH = HR and hence, if the evolution |ψ(t)〉 is governed by this H for some initial state |ψ(0)〉, then R|ψ(t)〉 is also a state evolving under H but for a different initial condition R|ψ(0)〉.

Let us consider how these symmetries affect the basis states, which are the initial states of interest to us. In terms of the basis states |a, b, S〉, they effect the following modifications:

$${{R}_{2}}:{\text{|}}a,b,S\rangle \mapsto {\text{|}}{\kern 1pt} a,b, - S\rangle ,$$
((7))
$${{R}_{1}}:{\text{|}}a,b,S\rangle \mapsto {\text{|}}{\kern 1pt} - {\kern 1pt} a,b, - S\rangle ,$$
((8))
$${{R}_{3}}:{\text{|}}a,b,S\rangle \mapsto {\text{|}}a, - b,S\rangle .$$
((9))

We can also apply combinations of Ri, and each operation can be applied or not, which provides together 2 × 2 × 2 = 8 possibilities. From a single initial state |a = 1, b = 1, S = 1〉 they produce all eight basis states. Hence, from our knowledge of the final state for one initial basis state we immediately obtain all eight results for the initial basis states, and hence the complete two-qubit gate.

As a result, we find that the braiding operation effects the following operator on the qubits and the coupler spin:

$$U = \exp \left[ { - i\frac{\pi }{4}{{S}^{y}}\tau _{1}^{z}\tau _{2}^{z}} \right]$$
((10))

in terms of the coupler-spin, Sy, and the Pauli matrices of the qubits. One can see that this operator commutes with the symmetry operations.

DISENTANGLING QUBITS FROM THE COUPLER SPIN

In the previous section, we found the result (10) of the two-qubit braiding. However, in addition to modifying the state of the pair of qubits, this operation entangles the qubits with the coupler spin (unless it is in an eigenstate of Sy). Since for the purposes of quantum computation, one needs to be able to perform two-qubit gates which do not affect other degrees of freedom, this feature is undesirable. We show below how this can be avoided, that is how one can implement a two-qubit gate, disentangled from the coupler spin.

This gate is based on the operation (10). The needed result can be achieved if at the beginning of the manipulations, both qubits are located in the same chain, and not on different sides of the coupler spin. We assume that both qubits are placed in chain 1. To implement the operation, the second qubit (which is closer to the T-junction) is first shifted to chain 3 through the junction and past the coupler. Then the braiding is performed as described above. Finally, the second qubit is returned from chain 3 back to chain 1.

To find the overall effect of this operation, using Eq. (10), we should conjugate it with the shift operation V for the second qubit (chain 1 ↔ chain 3). The latter is just the Sy-controlled flip operation on the second qubit, that is V = 1 for Sy = 1 and V = \(\tau _{2}^{x}\) for Sy = –1. Explicit conjugation for these two cases then gives us the resulting two-qubit gate:

$${{U}_{{2qb}}} = \exp \left[ { - i\frac{\pi }{4}\tau _{1}^{z}\tau _{2}^{z}} \right].$$
((11))

As one can see, this operation is indeed a two-qubit gate, disentangled from the coupler spin.

DISCUSSION

Our findings in the previous section imply that a two-qubit gate can be implemented without correlating the qubit states with the coupler spin. This method is particularly useful since it allows one to implement logical quantum gates directly on the qubits without extra manipulations with the coupler spin (cf. similar results for single-qubit gates in [12]). In particular, this gate is insensitive to the initial coupler state, even if the coupler is in a mixed state. Together with single-qubit operations, it can be used to construct a universal set of quantum gates for quantum algorithms. We note that the resulting two-qubit gate (11) is a Clifford operation.

Since Clifford operations are not sufficient for quantum computations, the latter observation implies that extra gates are needed. For this purpose one could use auxiliary manipulations, including manipulations of the coupler spin or longitudinal local magnetic fields. We remind that the considered qubits are not protected against the longitudinal noise, but as we see, this provides a possibility to perform needed logical gates.

Another comment is in order concerning the influence of noise. As we have seen, partial topological stability of the considered qubits manifests itself in protection against noise in the parameters of the Hamiltonian (1), but not against longitudinal local noise. Thus experiments with this kind of qubits need to rely upon corresponding long decoherence times of the constituent ordinary qubits. Depending on the realization of qubits, decoherence due to longitudinal and transverse noise may be dramatically different, and then this partial topological protection is especially useful for longer coherence of the (pseudo-) topological qubits.

Let us also discuss the results of Section 3. They demonstrate the robustness of the result of the manipulations on the qubit states in the following sense. First, the result of the operation on the state |a = 1, b = 1, S = 1〉 is robust (up to a phase factor) with respect to deviations of the parameters during the manipulations from the ideal path—as long as the manipulations are adiabatic. The same applies to each of the other seven basis states. If these eight phases are the same, the operation is well defined up to an overall phase factor, which is irrelevant. However, if these eight basis states acquire eight different phases, this changes the operation considerably. The results of Section 3 demonstrate that these phases are bound together and cannot differ.

Moreover, these results show that the description of the operations also holds for a wide range of non-ideal manipulations (that deviate from the ideal not only at stages 2 or 3 as in [15]). Furthermore, this also allows covering the cases of finite (non-infinite) freezing transverse fields, including situations where these fields are finite at the beginning of the operation or vary spontaneously in the process. All these results demonstrate (partial) topological stability of the qubit system.

Work at HSE was supported via its Basic Research Program.

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Correspondence to Yu. Makhlin.

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Contribution for the JETP special issue in honor of I. M. Khalatnikov’s 100th anniversary

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Makhlin, Y., Backens, S. & Shnirman, A. Disentangled Quantum Operation on Majorana Qubits in Qubit Chains. J. Exp. Theor. Phys. 129, 733–737 (2019). https://doi.org/10.1134/S1063776119100212

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