Decay rates for topological memories encoded with Majorana fermions
Recently there have been numerous proposals to create Majorana zero modes in solid state heterojunctions, superconducting wires and optical lattices. Putatively the information stored in qubits constructed from these modes is protected from various forms of decoherence. Here we present a generic method to study the effect of external perturbations on these modes. We focus on the case where there are no interactions between different Majorana modes either directly or through intermediary fermions. To quantify the rate of loss of the information stored in the Majorana modes we study the two-time correlators for qubits built from them. We analyze a generic gapped fermionic environment (bath) interacting via tunneling with different components of the qubit (different Majorana modes). We present examples with both static and dynamic perturbations (noise), and using our formalism we derive a rate of information loss, for Majorana memories, that depends on the spectral density of both the noise and the fermionic bath.
Topological quantum computation requires the existence of topologically ordered states whose low energy excitations follow non-Abelian statistics. The subspace of states corresponding to a fixed number of quasiparticles is degenerate, to an exponential precision, in the separation between quasiparticles, and an exchange of the positions of these anyonic excitations, also known as braiding, leads to a unitary transformation within this low energy subspace. These unitary operations are insensitive to the exact path used to perform the braiding operation and in many cases, for an appropriate encoding, braiding operations correspond to “standard” one- and two-qubit gates within the low energy subspace. These operations can be used as building blocks for fault tolerant quantum computation.
There are many candidate systems for experimental realizations of topological phases of matter with these properties. There is preliminary evidence that the fractional quantum Hall state may have non-Abelian excitations (1); (2); (3). Spin-triplet pairing superfluidity occurs in the A-phase of (4); (5) and in strontium ruthenates (6); (7); (8); (9); (10); (11), in which half quantum vortices would be non-Abelian (12); (13). There are also proposals to realize chiral p-wave superconductors in ultra-cold atom systems (14); (15); (16). Furthermore there have been many advances towards producing topological states of matter in layered heterojunction systems (17); (18); (19); (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (30).
Virtually all current experimentally viable proposals for platforms for topological quantum computation only support Ising type anyons which are carried by Majorana fermion modes. Colloquially speaking these fermions are half of a regular fermion. More precisely they are self-adjoint operators which can be written as a sum of an annihilation and creation operator for one fermion mode and which satisfy the algebra:
Any two Majorana fermion operators can be combined into a regular fermion mode and its adjoint via and .
The topological qubit is made up of four spin polarized MBSs and (31). These can be combined into two sets of creation and annihilation operators:
For the logical basis it is convenient to work in the even fermion parity subspace. The qubit basis can be chosen to be and where the ’s and ’s refer to the occupation numbers relative to the complex fermion operators in Eq. (2). Because of fermion parity conservation, any operation that does not entangle the states with the environment cannot mix even and odd fermion parity states for the qubits. As such, all gates acting on the topological qubit should not take the system out of the logical subspace. Furthermore all the operators of the single spin Clifford group may be produced by braiding the four vortices of our qubit leading to potentially topologically protected gates (32). In particular the various single qubit operations in our logic basis may be conveniently written in terms of the Majorana operators. For future use we note that in this encoding
Here all the sigma matrices are with respect to the logic basis and . We will primarily be interested in correlators of the form . We will proceed to calculate these below.
The Majorana operators are zero modes of some mean field Hamiltonian so it can be argued that these modes are protected from decoherence as the mean field Hamiltonian when restricted to the subspace generated by these modes is zero. One of the open tasks of topological quantum computation is associated with understanding the extent of this protection. This is the subject of this paper.
Ii Summary of main ideas
In this section we outline the setup of the rest of the paper. We present the relevant Hamiltonian and discuss its basic properties. We describe the type of qubit we will focus on in the text, a localized Majorana mode, and give an overview of some other encodings we shall not consider in this paper. We describe the kinds of calculations of memory coherence we are going to do in this paper. We also give a Section by Section outline.
We begin our discussion with relevant Hamiltonians. The Majorana fermions interact with the external environment via tunneling type Hamiltonians. On symmetry grounds, for a single Majorana mode, any such interaction may be written as:
Here is the localized mode function associated with the Majorana bound state, is any local bosonic field, which in the simplest case is a tunneling amplitude (complex number) and is a regular (complex) fermion field. In this paper we will analyze multiple Majorana fermions coupled to different types of environments via Hamiltonians of the form given in Eq. (4). Furthermore the fermions in the bath will always be assumed to be gapped, for example, electrons in an insulating or superconducting material (environments composed of gapless fermions, instead, would obviously lead to decoherence).
There are many examples of microscopic situations where Hamiltonians of the form given in Eq. (4) arise, one is as follows. If one writes the mode expansion of the electron creation and annihilation operators in the (superconducting) system of interest, one finds that:
Here stands for the eigenoperators of the BdG equations, with non-zero energies, while and are the components of the corresponding eigenmode of the BdG equations. is the Majorana fermion corresponding to the zero energy mode. Now consider an insulating substrate below a system which may be described by Eq. (5) above. A concrete example is given by the bulk of a topological insulator in tunneling contact with a superconductor as shown in Ref. (33). For a static Hamiltonian the bulk and surface states are orthogonalized, but dynamical effects such as phonons or two-level defect systems can alter the original Hamiltonian and turn on a hybridization. This perturbation takes the form of a tunneling between the electrons: , where controls the amplitude of fluctuations of the tunneling coupling. can be due to phonons, two-level systems, or even classical sources of noise. The electrons come from the insulating (gapped) system, which comprise the fermionic component of our bath. This illustrates one of the many ways to arrive at Hamiltonians of the form Eq. (4).
The coupling Hamiltonian that is derived in the paragraph above is local. The terms in Eq. (4) are local and couple to only one Majorana mode, with no long distance coupling between the modes of any form. In this paper we shall focus on sets of baths that couple to each Majorana individually. We would like to stress now and henceforth that even by coupling to individual modes, one at a time (with no cross mode coupling), the bath can be very damaging, in many cases leading to zero coherence for long times.
Below, we look at decoherence by analyzing qubit correlations such as , which, as we show in this paper, factorizes when the baths that couple to each Majorana are uncorrelated with one another:
Thus, even though the qubit is defined non-locally using spatially separated Majorana fermions, below we will show that the decay of the memory is controlled by the product of the two-time correlations of the separate Majorana modes. It then suffices to understand the effect of the bath on each Majorana fermion separately.
At this point its worthwhile to stress that the qubit encoding given above is not unique. A particularly interesting example of a different encoding, given by Akhmerov (37), is a fermion parity protected encoding. There, the qubit is made from fermion parity preserving operators:
that commute with both the tunneling Hamiltonian and the Hamiltonian for the environment. Here the are the operators in the mode expansion of the fermionic field in the bath ( here labels the mode, which can be momentum, for example). For a finite system, such as mid gap Carroli Matricon deGennes states in vortex cores, this compound qubit is very efficient. However we stress that, in the presence of a bath (say made by continuum states), the construction of an operator that is protected because of parity conservation requires a product of infinitely many operators: which is not practical or easily experimentally measurable. One could also truncate the product so as to account for a system, and the terms omitted are those assigned to the bath, as depicted in Fig. 1. In this case, however, because the operator lacks degrees of freedom assigned to the bath, parity can leak to the environment decohering the qubit. As such we will ignore all “compound” encodings for the rest of the paper.
Finally, we would also like to mention that the above scheme, with simple, non-compound, Majorana encoding, generalizes to multiple qubits. One possible encoding (though not the most economical) is to use four vortices and as such four Majorana modes per qubit. For this and any other encoding all possible correlators for the quantum memory may be expressed as expectation values of various products of Majorana operators(38). All quantum coherences for our qubits may then be computed by studying Majorana mode correlators which we study below.
In carrying out this program, we will analyze two distinct types of environments: the first is when couplings change suddenly but remain static thereafter, and the second when the environment changes dynamically. We show that that in the static environment case the tunneling Hamiltonian merely leads to a finite depletion of the Majorana two-time correlations. In this case, much of the information stored in these modes survives for arbitrarily long times.
More generally, for dynamic environments, we obtain an expression for the rate of loss of information stored in the Majorana operators that depends on the spectral density of the noise and of the fermionic bath. We present several examples of noise that can be studied essentially exactly, for instance classical telegraphic noise, as well as both classical and quantum Gaussian fluctuations.
The results in the paper are presented as follows:
In Section III we present general considerations involving the coherence properties of Majorana modes. We show that under reasonably generic initial conditions the coherence of the Majorana modes does not depend on their initial states. Furthermore we show that the two time correlation functions, coherences, factorize as a product over coherences for individual Majorana modes, that make up the quantum memory, interacting with their individual environments. As such we may reduce the problem of the coherence of the quantum memory to the problem of the coherence of one Majorana mode in tunneling contact with a (gapped) fermionic reservoir.
In Section IV we take a first step towards a calculation of the coherence of a single Majorana mode. We begin by describing the Keldysh technique relevant to Majorana modes. We present combinatorial tricks that make is possible to efficiently convert Keldysh computations using a mixture of Majorana and regular fermionic modes into a more familiar computation which uses only regular fermion modes. We then present an example where, for simplicity, we treat the fermions in the bath as free (non-relaxing approximation). We also present a general formula for the coherence of a Majorana qubit that is used several times in the remaining analysis.
In Section V we present several related classical models for the fluctuations of the bath. We solve these models essentially exactly, by mapping the problem of the coherence of a single Majorana mode to the problem of a particle undergoing classical diffusion. We use this technique to study classical fluctuations of the tunneling amplitudes and energy levels of the reservoir (we primarily focus on Gaussian fluctuations). In all cases we find decoherence with a rate that depends on the spectral density of the fluctuations in the reservoir. In many cases the decoherence due to an individual fermion mode has a power law time dependence but it will turn out that a bath made of many weakly interacting modes leads to exponential decay of coherence for intermediate times.
In Section VI we conclude. In light of the results we obtain in this paper, we critically examine the degree in which quantum memories can be encoded using Majorana fermions when these are in contact with a dynamical environment. We show that the coherence of the Majorana mode is controlled by the coherence of the bath it interacts with.
In Appendix A we compute exact dressed zero modes for static quadratic Hamiltonians, which we use to verify the validity of our results in Section IV. In Appendix B we present a rather technical calculation of a Majorana mode interacting with a fermionic bath with fully quantum mechanical Gaussian fluctuations. To leading order we find a decay similar to classical computations. In Appendix C we present various technical calculations, used throughout the rest of the text. In particular, in Appendix C.1 we show that our results are independent of coding subspace, in Appendix C.3 we present some technical arguments (which are used in Section V) in favor of weak (negligible) coupling of the fluctuation for the various fermionic modes. In the rest of the appendix we derive formulas used in the main text.
We begin with a study of the general properties of the dynamics of a system of Majorana modes. We will focus on a computation of correlators involving Majorana operators. This will allow us to study the coherence properties of a topological quantum memory which is based on qubits made up of localized zero energy modes. In this Section we will adhere to very general Hamiltonians and we will study only properties that are essentially independent of the form of this Hamiltonian. This will set us up for studies of specific types of Hamiltonians in Section IV. From the outset, we would like to specify the initial conditions or equivalently the density matrix when the system is initialized at . We will assume that initially the density matrix factorizes into a product of the form:
Here is the density matrix for the entire system, while represents and arbitrary non-equilibrium density matrix for the Majorana modes. The are arbitrary, not-necessarily equilibrium, density matrices for the environments of the individual Majorana modes. No specific “ensemble” is assumed. This form is a reasonable, consistent assumption for the initial states of system plus bath, particularly so, as many experimental methods of initialization produce such states.
For our qubit memory persistence between times and is captured by the two-time correlators such as . We note that, because the initial, , state breaks time-translation invariance, generically these correlators are functions of both and . Here we shall focus specifically on correlations, like , between the state prepared at and the state at a later time time which characterize the degree to which the information encoded in the qubit at the initial time survives interaction with the bath when it is retrieved at a later time .
The key results of this section, which are used repeatedly later in the text, may be summarized by saying that even though the factorization form given in Eq. (8) does not survive Hamiltonian evolution the expectation values of various correlators like or equivalently products of Majorana fermions, to be defined precisely in Eqs. (12) and (13) below, do factorize into products of expectation values for individual Majorana modes. This factorization survives for arbitrary times.
iii.1 General ideas
We will consider a set of Majorana modes each interacting with its own fermionic environment, see Eq. (8). We will see that there is decoherence even without direct interactions between different Majorana modes or between their respective environments. One can show that, in the limit when the spatial separation between the Majorana modes is large, the case when multiple Majorana modes interact with a common fermionic bath reduces to the case of uncorrelated non-interacting baths (see Appendix C.2). The Hamiltonian pertinent to each mode may be written as:
Here are some bosonic modes and labels the Majorana modes. The total Hamiltonian is given by . We will be interested in correlators of the form . Here all operators are in the Heisenberg picture, and is given by
where and stand for time-ordered and anti-time-ordered products, respectively. Notice that at all times.
Now, by Taylor expanding the time-ordered and anti-time-ordered exponentials in Eq. (10), taking various commutators, grouping terms and using the fact that , we may write that
with and having no factors of . Because must be fermionic (this can be seen from the fact that the Hamiltonian and all its powers are bosonic) we may deduce that and are, respectively, bosonic and fermionic operators. By the conservation of fermion parity we know that the expectation value of any operator . Finally, because is Hermitian, it also follows from the properties above that and are Hermitian as well.
Now, it follows that
where we used going from the first to the second line of Eq. (12) that the environments and the Majorana states are initially disentangled so expectation values factorize. Note that this comes about because in the Heisenberg picture the expectation values for operators are taken with respect to the initial state, at . For the third line we have used that the expectation value of any fermionic operator should be zero. Note that because is Hermitian this implies that .
The following factorization formula can be similarly showed:
for distinct , . To show this expression, one uses Eq. (11) and again that the expectation values are computed with respect to the initial density matrix given in Eq. (8) which has the property that the environments are uncorrelated with each other and with the initial Majorana states. We see that this factorization formula is independent of the initial state of the density matrix of the bath. As such our formalism captures highly non-equilibrium initial conditions.
iii.2 Qubit memory correlations
The degree of persistence of memories assembled using Majorana fermions can be quantified by the correlation between the qubit state, encoded as in Eq. (3), at two times :
Notice that the factorization implies that, even though the qubit is defined non-locally using two spatially separated Majorana fermions, the decay of the memory is controlled by the product of the two-time correlations of the two separate Majorana modes. In particular, the decoherence rate is independent of the initial state of the quantum memory (that is correlators of the form do not enter the result).
Thus in the case of uncoupled well separated Majorana modes each interacting with its own environment the task of determining the persistence of topological quantum memories based on Majorana fermions is reduced to the calculation of the coherences in the presence of different fermionic environments. We carry out this program henceforth.
Iv Keldysh calculation of coherence
We now proceed to describe the technical details associated with studying dynamics. For generality and later use we will study both static and time dependent Hamiltonians. Based on the discussion given in Section III for the purposes of computing coherences it will be sufficient to focus on a single Majorana mode. As such we will drop the subscript , see Eq. (9), henceforth.
iv.1 General Observations
We will convert the computation of the Majorana correlations into a Keldysh calculation carried out using only the bosons and regular complex fermions inside the reservoir. (For a review of standard Keldysh techniques see e.g. (39); (40); (41).) We will calculate the following correlator:
Here the expectation value is taken relative to the density matrix at while and stand for time ordering and time antiordering respectively. To make the computations tractable we will assume that . Here is any initial density matrix acting on the subspace of the Majorana modes while is the thermal density matrix for the regular fermion modes.
To compute the correlator in Eq. (15), we will use Eq. (9) and work in the interaction picture with respect to the rest of the Hamiltonian . We will expand the ordered exponentials in powers of and collect and contract all the s to eliminate them. In what follows will show that
where , and stands for the Keldysh ordering that combines the forward and backward propagation, and the index labels the two pieces (forward and backward) of the ordered product. (Notice though that the operator in the exponential comes with the same sign in the and products.)
Below we give the essential arguments needed to derive Eq. (16). To carry out this program, let us introduce a short-hand notation . Now expand Eq. (15) in powers of , and focus on the term with insertions, with from the expansion of the -ordered exponential and from that of the -ordered exponential. By fermion parity conservation and using our assumption that the system-bath initial density matrix is factorized we know that is even. The insertions of our interaction Hamiltonian are of the form
We show in curly brackets the modes at and at , to help single them out for constructing the argument below. Our strategy to convert this calculation to a “regular” Keldysh calculation will be to move the Majorana modes ( terms), including the at , by taking appropriate commutators, till they are all at the left hand side, adjacent to the inserted at . We will move along the contour ordering direction (see Fig. 2). We will then use the relation to eliminate these modes altogether. All that remains is a computation of the commutators. Because of the form of the Hamiltonian, computing commutators is equivalent to computing an overall sign for the term in the expansion. By noting that the Hamiltonian is bosonic we obtain that the overall sign is only due to the anti-commutation of the ’s with the and inside the terms. We shall move each mode to the very left in two steps: we first move the mode at to the very left towards ; then we move all the remaining modes there as well.
In the first part of the procedure is to obtain the contribution of the Majorana fermion inserted at . We note that the number of signs it picks up depends on its position along the contour relative to the other modes it picks up one sign for very mode it passes so there is an overall sign of .
Now for the rest working from left to right, the first Majorana mode that needs to be moved picks up no signs as it does not pass over a term, but the second picks up one sign as it passes over one such term. Similarly, the third picks up two signs, and so forth. Finally the th Majorana mode (last to be moved, sitting all the way to the right) picks up factors of . The product of these factors yields .
These are precisely the terms that appear in the series expansion of Eq. (16), and therefore we can continue the calculation utilizing this expression. We should point out that for complex fermions coming from Majorana insertion corresponds to literal ordering on the Keldysh contour, without any fermionic minus signs, because the original Hamiltonian was bosonic [this can also be seen step-by-step in going from Eq. (17) to Eq. (18)]. This fact leads to the modified sign for the fermionic -ordering:
Now, we turn our attention to the computation of Eq. (16). We do so in steps, computing the expectation values by first tracing the fermions () and then subsequently tracing the bosonic degrees of freedom. Even in the case where there are interactions for the fermions, we can still treat the theory as quadratic in the fermions and include the interactions (with photons or phonons) as a coupling of the fermionic bilinears with the mediating bosons, which we label by . Alternatively, we may think of the fields fields as Hubbard-Stratonovich decoupling fields(42).
We can thus write
We remind the reader that all functional integrals are along the Keldysh contour. The action is that of the interaction mediator field and contains the dressing from the integration of the fermions, which are integrated out first as explained above. The normalization is
This procedure works because it possible to calculate partition functions, Green’s functions, integrate fields out etc. along any contour, in particular along the Keldysh contour as used here. We then express the fermionic correlators in terms of their Green’s function,
where the and are, respectively, the electron and hole fermionic Green’s function, and we have used the fact that the bosonic fields can be treated as c-numbers as they are inside the bosonic path integral. As stated previously and are slightly unusual Green’s functions, with no fermionic minus signs (only plus signs), as shown in Eq. (19). Let us define , so we can then write
We remark that the expression in Eq. (23) was derived without any approximations. It holds for interacting electrons as well, as long as the interactions are included via an external bosonic field denoted by above. Furthermore we would like to note that though it is not used anywhere in this paper, but a similar path integral formulation using Grassmann variables may be done without any decoupling fields, for regular quartic fermionic interactions. A systematic Keldysh diagrammatic perturbation theory may be derived from it.
Here , refer to time ordering and time anti-ordering operators. This form places the time ordering or antiordering terms () with the appropriate fermion correlators so it can be used directly in calculations without having to use a path integral. The factor of two going from the first to the second line comes from a symmetry (which also allowed us to simplify Eq. (24) above to contain six rather then twelve terms). Because of exponentiation of disconnected diagrams, if we can safely ignore higher order correlations among the ’s, we may write that:
A quick way to derive the extra factor of in Eq. (25) above is by noting that it is a symmetry factor associated with the ability to permute the two Majorana insertions without changing the diagram [alternatively we can do a combinatorial check, or use Eq. (23)].
iv.2 Simple examples
Let us consider simple cases where the are simply constants , switched on at . In this case the expression in Eq. (23) simplifies to
where , with and exact 2-point electron and hole Keldysh propagators, including the effects of interactions. To be explicit at this level of approximation our formalism handles all the dynamics of the fields but treats fermionic interactions to quadratic order. The stand for terms of order that involve the 4-point Green’s functions . We shall not do so in this paper, but by including these and higher terms it is possible to handle all fermionic interactions as well.
Taking into account all the four cases in the sum over top and bottom insertions , one can write
We now consider a case where this formula will be particularly useful. We Consider the case when the bath is described by the Hamiltonian
In this case we have
If the bath has energy eigenenergies away from zero energy (i.e., there is a gap ), we may drop the oscillating terms in the limit of , so we can write
In this case, the Majorana memory decays to independent plateaus at large times. Thus, as long as the sum converges, the memory is retained to a finite extent. This result is confirmed by a time-independent re-diagonalization in the presence of the , which is shown explicitly in Appendix A where a new exact zero mode is calculated. Here we simply note that the finite depletion found in this case is a simple consequence of the fact that the modes change once the coupling is switched on. Also, we compute the sum , and find it to be finite, for a specific tunneling model in Appendix C.4.4.
V Fluctuating Hamiltonians
So far we have studied static Hamiltonians. To gain further insight it is interesting to extend our results to fluctuating couplings (which may come from time dependent classical fluctuations or from quantum dynamics). We shall focus on three cases, in all three the fermionic action is quadratic. In the first case we study we consider the situation when the are simply replaced by classical variables , like we did in Sec. IV.2, but now they depend on time. The second case is that when the energies of the electrons in the bath fluctuate in time, because of environmental fluctuations. The third case is a generalization of the first one, where we treat the quantum mechanically with their fluctuations governed by a quadratic action. We treat the first two cases here, and the third, more technical one, in Appendix B.
In the first two cases, one can generalize the expression in Eq. (30) simply by taking or :
and then average over statistical fluctuations of the and .
The computation of the Majorana correlations can be greatly simplified as follows. Notice that, for each mode , the argument in the exponential in Eq. (33) can be viewed as the magnitude square of the position of a particle moving in two-dimensions, or alternatively the modulus square of a complex number moving on the plane:
Below we will argue both in the cases of fluctuating amplitudes and energies that the probability distribution for the “position” is Gaussian:
with the time-dependent width of the distribution, which we will compute below for each case. With this Gaussian distribution for the , we can compute the average Majorana correlation,
In the last step we assumed that there are many modes in the fermionic bath, each making a small contribution (or order inverse volume) so we may re-exponentiate the product. The examples below are studied using this expression.
v.1 Fluctuating amplitudes
The fluctuations of the are assumed to be Gaussian distributed according to