Physical Implementation of a Majorana Fermion Surface Code for
Fault-Tolerant Quantum Computation
We propose a physical realization of a commuting Hamiltonian of interacting Majorana fermions realizing topological order, using an array of Josephson-coupled topological superconductor islands. The required multi-body interaction Hamiltonian is naturally generated by a combination of charging energy induced quantum phase-slips on the superconducting islands and electron tunneling. Our setup improves on a recent proposal for implementing a Majorana fermion surface code Maj_Surf_Code (), a ‘hybrid’ approach to fault-tolerant quantum computation that combines (1) the engineering of a stabilizer Hamiltonian with a topologically ordered ground state with (2) projective stabilizer measurements to implement error correction and a universal set of logical gates. Our hybrid strategy has advantages over the traditional surface code architecture in error suppression and single-step stabilizer measurements, and is widely applicable to implementing stabilizer codes for quantum computation.
Superconducting qubits based on Josephson junctions provide a promising platform for the implementation of a large-scale, fault-tolerant quantum computer Girvin (). The key physical degree of freedom that defines a superconducting qubit is the phase difference of the order parameter across a Josephson junction. In recent years, remarkable progress has been made on coherence times, high-fidelity gate operations and read-out protocols for superconducting qubits devoret (); martinis (); ibm ().
In addition to having a phase degree of freedom, superconductors host fermionic quasiparticles. In particular, quasiparticles with energies inside the superconducting gap can be localized in superconducting weak links or vortices. The quasiparticle occupation number could be exploited as an internal degree of freedom to encode information in a Fock space, opening up the possibility of fermionic quantum computation. Of particular interest are Majorana fermions, excitations which arise as a spatially-localized “half” of a zero-energy fermionic quasiparticle in a topological superconductor kitaev (); Read_Green ().
Several important considerations motivate us to study schemes for robust quantum computation using fermionic quasiparticles in superconductors. From a quantum information perspective, it has been theoretically shown that fermions have more computational power than bosons Bravyi_Kitaev (). Furthermore, a fermionic quantum computer can directly and efficiently simulate many-body Fermi systems with local interactions, without the computational cost of mapping to non-local bosonic systems.
In this work, we propose a physical implementation of a quantum error-correcting code using an array of Majorana fermions as the underlying physical degrees of freedom, which was recently introduced and termed a Majorana fermion surface code Maj_Surf_Code (). Unlike the conventional surface code with bosonic physical qubits Surf_Code_I (); Surf_Code_II (); Surf_Code_III (); Surf_Code_IV (); Surf_Code_V (), the Majorana fermion surface code is based on commuting Hamiltonians of interacting Majorana fermions in two dimensions that exhibit topological order Maj_Surf_Code (); Xu_Fu (); Terhal (); Nussinov (). Importantly, our proposal combines the engineering of the static parent Hamiltonian with projective measurements for active error correction and universal gate implementation. Our Hamiltonian-measurement hybrid strategy has significant advantages over a measurement-only approach as employed in the conventional surface code, and provides a general approach to implementing stabilizer codes.
Our hybrid strategy is particularly adapted to providing the error correction capability that is required for a scalable quantum computing architecture. For any quantum computer operating at non-zero temperature, a finite density of spurious excitations will be thermally excited, interfering with encoded information and leading to the failure of fault-tolerance in the absence of active error correction. As a result, Hamiltonian-only approaches to quantum computation, including those based on non-Abelian topological phases, will be unable to proceed correctly in the absence of measurement-based error correction.
In contrast, our Hamiltonian-measurement hybrid approach achieves scalability and fault-tolerance in two ways. First, engineering an ideal stabilizer Hamiltonian naturally provides error suppression at temperatures below the energy gap, and remarkably permits single-step measurements of multi-body stabilizer operators, which minimizes readout errors. Furthermore, as the interactions between the physical degrees of freedom are described by the stabilizer Hamiltonian in our setup, we are able to take non-idealities in stabilizer measurements into account in a controlled fashion. Second, the constant projective measurements of stabilizers can detect unwanted excitations (i.e., stabilizer flips) that do occur. By collecting measurements at several time-steps, an error can be optimally decoded and used to correct a logical qubit during readout as in the conventional surface code, leading to a scalable quantum computing architecture Surf_Code_V (); Surf_Code_Error ().
This paper is organized as follows. First, we introduce a commuting Hamiltonian of interacting Majorana fermions that exhibits a topological order of Fermi systems, as thoroughly described in Maj_Surf_Code (). Next, we demonstrate that this exact Hamiltonian can be realized in an array of Josephson-coupled topological superconductors, and projective measurements of the commuting stabilizers can be achieved using current superconducting qubit technology. This combination of engineering a static Hamiltonian and performing constant measurements constitutes a full implementation of a Majorana fermion surface code. Potential advantages of our implementation over previous proposals Maj_Surf_Code (); Terhal () are discussed. For the sake of completeness, we also briefly review how logical qubits may be encoded and manipulated to achieve a universal set of gates, as detailed in Maj_Surf_Code ().
Ii Fermionic Topological Order
We begin by introducing a commuting Hamiltonian that realizes a fermionic topological order. Consider a square-octagon lattice with one Majorana fermion per lattice site, as shown in Figure 1a. The Hamiltonian is defined as
where is the product of the four Majorana fermions bordering a square plaquette , while is the product of the eight Majoranas around an octagonal plaquette . Any pair of Majorana fermions () anti-commute and satisfy . Therefore, each of the operators appearing in the Hamiltonian squares to the identity, and has eigenvalues . Furthermore, as any pair of operators in the Hamiltonian overlap on an even number of Majorana fermions, all terms in the Hamiltonian mutually commute, and constitute a complete set of stabilizer operators. The ground state satisfies and for all plaquettes , .
The Hamiltonian (1) belongs to a family of solvable “Majorana plaquette models” in two dimensions Maj_Surf_Code (); Bravyi (), and arises as a low-energy effective Hamiltonian studied in the work of Xu and Fu Xu_Fu () and others Terhal (); Nussinov (). The ground state exhibits four-fold topological degeneracy on the torus, and realizes a topological order of Fermi systems. Topological excitations are created from the ground state by acting with the product of Majorana fermions along lines; this flips the eigenvalues of plaquette operators at the ends of the lines, creating a pair of topological excitations. The three fundamental excitations – labeled , , or according to the plaquette type defined in Figure 1 – may only be created in pairs and cannot be deformed into each other. These three types of excitations are bosons, but have mutual statistics relative to each other; see Maj_Surf_Code () for a detailed discussion.
We propose an exact physical realization of the Hamiltonian (1) using an array of square and octagonal islands of two-dimensional (2D) topological superconductors, which can be realized by coupling -wave superconductors to topological insulator (TI) surface states Fu_Kane (), or spin-orbit-coupled 2D electron gases 2DEG1 (); 2DEG2 (). By applying magnetic flux or current bias, we fix the phases of the superconducting islands in the configuration shown in Figure 2a, so that the phases of any three neighboring islands wind by around each tri-junction, leading to a 2D array of Josephson vortices. As demonstrated by Fu and Kane Fu_Kane (), a phase vortex or anti-vortex in the proximity-induced superconductivity on the TI surface binds a localized Majorana zero mode. As a result, the phase configuration of the superconducting islands shown in Figure 2a produces the desired square-octagon lattice of Majorana fermions.
Iii Physical Realization of the Stabilizer Hamiltonian
We now engineer the 4-body and 8-body interactions in the Hamiltonian (1). First, we show that the 4-body interaction is naturally generated by the charging energy on the square superconducting islands. Our analysis closely follows Ref. Maj_Surf_Code (), where it was first demonstrated that the effect of superconducting phase-fluctuations on Majorana zero modes bound to vortex cores may be used to engineer non-local, multi-fermion interactions. Let us consider the effect of a small charging energy on a single square island , holding the phases of all other islands fixed. The Hamiltonian for this island is given by
The first term describes the Coulomb energy due to the excess charge on an island, which is capacitively coupled to ground and to the other islands in the array. Here is an offset charge that can be continuously tuned by a gate voltage. The second term describes the Josephson coupling of neighboring islands with energy , with tuned so as to produce the superconducting phase configuration in Figure 2a. Importantly, as the charging energy and the Josephson energy do not commute, the superconducting phase is a quantum-mechanical variable.
A small charging energy induces quantum fluctuations of the superconducting phase on the square island. Small fluctuations result in a harmonic oscillator spectrum with level-spacing . As originally found in Maj_Surf_Code (), a remarkable feature of our setup is that quantum phase slips permute vortices at tri-junctions and hence the Majorana fermions bound to these vortices. Specifically, a phase slip on a square island exchanges the two Majorana fermions bound to the (anti-)vortices at diagonally opposite vertices in a (counter-)clockwise manner, via a sequence of intermediate configurations shown in Figures 3a-3d. The net effect of this phase slip is to permute the Majorana fermions as follows
The braiding of Majorana fermions in charing-energy induced quantum phase slip processes is crucial to our implementation of the Majorana fermion surface code and enables a new way of detecting the non-Abelian statistics of Majorana zero modes forthcoming ().
As shown in Maj_Surf_Code (), the low-energy effective Hamiltonian for the island in the limit takes the form of a tight-binding Hamiltonian describing a “phase” particle that tunnels between the minima of the periodic Josephson potential, located at . For a phase-slip, the contribution to the effective Hamiltonian is given by
Here is the on-site energy of the lowest harmonic oscillator state associated with each potential minimum, while is the tunneling amplitude between nearest neighbors, given by the amplitude of a phase slip . The offset charge produces a Berry phase in the effective Hamiltonian. Importantly, in our setup the phase particle carries an internal -dimensional Hilbert space resulting from the four Majorana fermions at the vertices. Here is the unitary operator that implements the braiding of Majorana fermions in a phase slip process, as described in (3), and is given by
where we have eliminated the phase factor of by shifting by a constant. From now on, we set to integer values by tuning the gate voltage, so that is exactly the -body interaction on square plaquettes needed for the Majorana plaquette model (1). We further require the size of each square island to be sufficiently large so that the four Majorana fermions at its corners have negligible wavefunction overlap, thus preventing any unwanted bilinear coupling within square islands.
To further implement the 8-body interaction on the octagonal islands, we consider the electron tunneling between adjacent square islands via nearest-neighbor Majorana fermion zero modes at tri-junctions. Such tunneling processes are described by Majorana bilinear operators such as in Figure 2. The tunneling amplitude depends exponentially on the the wavefunction overlap between and , which can be tuned by adjusting the distance between adjacent square islands and other device parameters (e.g., the chemical potential). We work in the regime , and treat the effect of perturbatively. A single-electron tunneling event flips the fermion parity of two adjacent square islands and leads to an excited state with a large energy cost of . The lowest order process that brings the system back to the ground state manifold of consists of four single-electron tunneling events between four pairs of nearest-neighbor Majorana fermions, such as shown in Figure 2. From fourth-order perturbation theory, the effective Hamiltonian for an octagonal island is found to be a ring-exchange term:
where is the product of the 8 Majorana fermions on the vertices of an octagonal island . The sum of the 4-body interaction (III) and the 8-body interaction (7) in our setup of the topological superconductor array precisely yields the commuting Hamiltonian (1), whose ground states and excitations will hereafter be used to encode quantum information, leading to a Majorana fermion surface code.
Physical realizations of Majorana plaquette models have been previously proposed in other platforms for Majorana fermions. In Ref. Xu_Fu (), Xu and Fu introduced a square-octagon lattice of Majorana fermions in an array of quantum spin Hall insulators and superconductors, and derived the Hamiltonian (1) when charging energy on the square superconducting islands and electron tunneling between the islands are present. The same approach was later employed to implement (1) in a network of semiconductor nanowires Terhal (). In these works, Majorana fermions sit at the interface between superconducting and insulating regions, rather than being bound to vortices as in our proposal—a crucial feature for implementing the stabilizer measurements in our Majorana fermion surface code, as discussed below. In our recent work with Hsieh Maj_Surf_Code (), a physical realization of a Majorana plaquette model in an array of hexagonal topological superconductor islands was studied. However, it was found that charging energy generates unwanted two-body interactions, in addition to the desired 6-body interactions between Majorana fermions. As we have shown above, this work improves on Ref. Maj_Surf_Code () and achieves an exact realization of the Majorana plaquette model (1) in an array of 2D topological superconductor islands.
Iv Measurement Protocol and Universal Quantum Computation
As proposed in Ref. Maj_Surf_Code (), we may use this physical realization of a fermion model with topological order as a platform for universal quantum computation – a Majorana fermion surface code. The ground state of the Hamiltonian is a highly-entangled many-body state and serves as a reference code state. Thermal excitations at non-zero temperature are stabilizer flips, and can only be created in pairs on the same types of plaquettes. However, even at low temperatures, the number of flips is given by , where is the minimum energy cost for a stabilizer flip and is the total number of stabilizers in the system. Therefore, despite being suppressed by the Boltzmann factor, the number of stabilizer flips scales extensively with the system size at a fixed temperature, so that any reliable, large-scale quantum computation necessarily requires active quantum error correction. To detect and handle errors in the Majorana fermion surface code, we perform constant measurements of all commuting stabilizers, except those used to encode logical qubits (see below), to project the system onto a stabilizer eigenstate. The measured stabilizer eigenvalues are recorded and used to correct errors, as detailed in Ref. Maj_Surf_Code () and references therein.
Logical qubits are encoded by stopping the measurement of two stabilizers of the same type (, or ) in subsequent cycles, so that the presence or absence of an anyon (= stabilizer flip) in the resulting “hole” defines the two states of the encoded qubit. A set of gates required for universal quantum computation (CNOT, Hadamard, and ) may be implemented by performing a sequence of measurements that have the effect of moving the logical qubits. A detailed description of gate and measurement protocols for the Majorana surface code is provided in Maj_Surf_Code (). Notably, since lattice symmetries exchange the and anyons, the Hadamard gate has a much simpler implementation than in the ordinary surface code Maj_Surf_Code ().
We now describe how to perform projective measurements of the commuting stabilizers in our setup. As discussed in Maj_Surf_Code (), the eigenvalue of a four-Majorana stabilizer can be determined by exciting the square island with microwave photons and measuring the energy gap to the next harmonic oscillator level Hassler (); Glazman (). As the sign of associated with the stabilizer operator in the Hamiltonian alternates between consecutive harmonic oscillator levels, the gap depends sensitively on the stabilizer eigenvalue. Therefore, a measurement of the energy gap directly determines .
Our implementation of the eight-Majorana stabilizer measurement deserves special attention. To measure the eigenvalue of , we increase the charging energy on an octagonal superconducting island to activate quantum phase-slips. Importantly, we require that the charging energy is small relative to the energy cost for flipping any stabilizer eigenvalue, and the charging energy is increased sufficiently slowly, on a time-scale larger than the inverse gap . Under this adiabatic condition, transitions to excited stabilizer eigenstates are avoided.
As in the case of square islands, a small charging energy on an octagon island induces quantum phase slips that permutes the eight Majorana zero modes at the vertices. As shown in Figure 4a-4d, a phase slip has the effect of permuting the Majorana zero modes as follows:
This transformation is implemented by the unitary operator where
In the presence of a charging energy, the effective Hamiltonian for an octagonal island acquires terms due to phase-slip events , in addition to the ring-exchange term (7) from single-electron tunneling. When the gate charge on the octagonal island is an integer, the effective Hamiltonian, including the contribution from the dominant phase slips, takes the form:
up to an overall constant. Here, is the amplitude for a phase-slip on octagonal island , and is a sum of quartic Majorana operators. Note that is time-dependent; as a function of time, increases as the charging energy for the octagonal island is slowly turned on during the stabilizer measurement and decreases as the charging energy is slowly turned off after the measurement has been completed.
As illustrated in Figure 5, we now measure the eight-Majorana stabilizer during the interval by coupling the octagonal island to a resonator and measuring the energy gap to the next harmonic oscillator level. For the Majorana surface code, the computational states consist of the eigenstates of the stabilizer Hamiltonian (1) that we engineered using an array of topological superconductor islands. The Hamiltonian for the octagonal island is no longer ideal (13) and includes an additional non-commuting term . However, as the Hamiltonian is gapped throughout the measurement process and the charging energy was turned on slowly, the many-body state of the system evolves adiabatically. When , a measurement of the energy gap, when performed for a sufficiently long duration, will determine the eigenvalue of , as in the case of square islands. After the measurement, we decrease the charging energy on the octagonal island sufficiently slowly, on a time-scale , so that the many-body state of the system evolves adiabatically back into an energy eigenstate of the ideal stabilizer Hamiltonian with as determined by the projective measurement.
We emphasize that this adiabatic measurement protocol is made possible by our Hamiltonian-measurement hybrid approach. The physical Hamiltonian for our square-octagon lattice of Majorana fermions is precisely the ideal Hamiltonian (1) whose eigenstates define the set of computational states. Therefore, even if stabilizer measurements are imperfect, the many-body state of the system will, after a sufficient interval of time, return to the computational basis with the desired stabilizer eigenvalue. This is in contrast to “measurement-only” approaches such as the ordinary surface code, where measurement errors cannot be handled in a controlled manner, as physical qubits are only coupled by measurements instead of a static interaction Hamiltonian.
In summary, we have improved on a recent proposal of implementing a Majorana fermion surface code in an array of topological superconductor islands Maj_Surf_Code (). Our proposal combines (1) the engineering a static stabilizer Hamiltonian by using the charging energy of superconducting islands and electron tunneling between islands with (2) single-step projective measurements for quantum error correction and gate operations. Given the rapid experimental progress towards the identification of Majorana fermions delft (); weizmann (); princeton (); jia () and the tantalizing prospect of integration with superconducting qubits marcus (), we hope the implementation of a Majorana fermion surface code proposed in this work be pursued. Recently, we have learned of an attractive proposal on a different implementation of the Majorana fermion surface code in an array of semiconductor nanowire based topological superconductors, taking a novel approach to stabilizer measurements Altland (). We also note that lattice systems of interacting Majorana fermion may host a wide variety of intriguing quantum phenomena including symmetry breaking franz (), topological order teo (); sau (), and beyond vijay (), continuing to inspire new approaches to quantum computation. Broadly speaking, the Hamiltonian-measurement hybrid approach introduced in our work may be widely applicable to implementing stabilizer codes for large-scale, fault-tolerant quantum computation and is likely to be advantageous over Hamiltonian-only or measurement-only approaches.
Acknowledgements.This work was supported by the Packard Foundation (LF), and the DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-SC0010526 (SV).
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