Quantum Computing: the Majorana Fermion Solution. By: Ryan Sinclair. Physics 642 4/28/2016

Transcription

1 Quantum Computing: the Majorana Fermion Solution By: Ryan Sinclair Physics 642 4/28/2016

2 Quantum Computation: The Majorana Fermion Solution Since the introduction of the Torpedo Data Computer during World War II, American society has gradually become dependent upon digital computers in their everyday lives. As the size of computers began to shrink, their integration into the average household expanded. Nowadays, computers are everywhere phones, calculators, watches, toys, and more to the point where they are almost taken for granted. Yet as we continue perfecting the classical computer, a new frontier has captivated the scientific imagination. While classical computers have vast potential, quantum computers are able to surpass their abilities by utilizing entangled bits as well as quantum mechanical properties such as quantized spins. There are several different methods for implementing the quantum bits (qubits) needed for quantum computing, and one promising solution builds these qubits from particles known as Majorana Fermions (MFs). Majorana Fermions are defined by the particle being its own antiparticle [1]. Currently, the most promising candidate for a natural MF is the neutrino/antineutrino, but this property has yet to be confirmed. Luckily, the scientific community is not completely reliant on nature. One can strategically manipulate natural processes in order to construct exotic quasiparticle excitations. The current strategy involves electron and hole excitations (Bogoliubov quasiparticles) that exist in superconductors because their creation and annihilation operators are related by γ(e) = γ (-E). As a result, γ = γ at the Fermi level (E = 0) which causes the particle (electron) and antiparticle (hole) to have eigenstates that are charge-neutral. Unfortunately, the problem is slightly more complicated. Spin degeneracy must also be broken in order to realize an unpaired MF. This condition requires the involvement of unique superconductors; in particular, unconventional superconductors which only involve a single spin band show promise. Currently, the spin-triplet, p-wave pairing and the spin-singlet, s-wave in combination with a topological insulator show the most potential to realize MFs, but the s-wave option produces MFs that are less affected by disorder. [1] In order to be used as a qubit, the MF must be bound to a topological defect, forming what is known as an Ising anyon. [1] Unlike free MFs, Ising anyons are described using non-

3 Abelian statistics, i.e. the operators describing this system are not commutative which distinguishes them from Fermions and Bosons. This results from the evolution of the system being described by a unitary matrix Ψ U Ψ. As matrix multiplication is non-commutative, the transformations above must also be noncommutative. Figure 1: Top view of a 2D topological insulator, contacted at the edge by two superconducting electrodes separated by a magnetic tunnel junction. A pair of MFs is bound by the superconducting and magnetic gaps. The tunnel splitting of the bound states depends on the superconducting phase difference φ, as indicated in the plot. The crossing of the levels at φ = π is protected by quasiparticle parity conservation. [1] These bound states that make up the qubit can be created using a 2D topological insulator contacting two superconducting electrodes at the edge which are separated by a magnetic tunneling junction. As shown in Fig. 1, the qubit s two states 0 and 1 are formed by two zero modes coupled by tunneling. Each of these states can be distinguished by the presence or absence of an unpaired quasiparticle. A qubit in this configuration is immune to both classical

4 and quantum errors assuming that the fermionic sites are separated by a medium with a gap in the excitation spectrum. [2] The qubit avoids dephasing through the two real MF operators γ1 and γ2 in combination with a complex Dirac fermion operator a = ½(γ1 + γ2). This requires a joint measurement of the MFs to determine the quasiparticle parity a a. Consequently, two MFs encode one qubit, so it takes 2n MFs to encode the quantum information in 2 n nearly degenerate states. [1] While theoretical error immunity gives the community hope that this system can be realized, all systems are exposed to noise effects which introduce errors into the system. Noise effects have been studied for hybrid topological systems using MF qubits with good results [3]. With a noise threshold of 0.85% for entangling operators, a high quality logical qubit can be constructed with less than one hundred MF qubits given a feasible error rate near 0.1%. This reduces the overhead in comparison to normal qubits by an order of magnitude as long as the charge tunneling events are suppressed to a very low level. [3] Now that is has been shown that MF qubits are a practical approach to supercomputing, one must figure out exactly how to construct the same universal logic gates that classic computers rely on. Here we utilize the non-abelian statistics exhibited by Ising anyons for topological quantum computation in which logical qubits are encoded in the topologically degenerate states of non-abelian anyons, and qubit operations are performed by braiding. [3] Braiding is the adiabatic interchange of two Majorana bound states, and it is described by a nonabelian unitary transformation Ψ exp(iπσz/4) Ψ where the Pauli matrix acts on the qubit formed by the two interchanging MFs. This operation is defined as topological because it is fully determined by the topology of the braiding which results in the exponent coefficient of exactly π/4. Unfortunately, braiding alone is insufficient for universal quantum computation. [1] The inadequacy of braiding requires a different approach in order to perform universal supercomputing. As a result, surface codes have been devised to provide an alternative solution. [1] In this scheme, the necessary logical gates are realized through sequences of measurements that move and braid the logical qubits. Logical qubits are created by ceasing the measurement of certain stabilizers in order to create holes with different possible anyon charges depending on the degrees of freedom that define a logical qubit. Logical qubits are composed of physical qubits which are defined by two energy levels that arise from the quantization of number/phase

5 fluctuations in a conventional Cooper pair box. The required measurements involve non-trivial commuting operators known as stabilizers and project onto code states which are highly entangled. [4] To achieve universal quantum computing, surface code involving bosonic degrees of freedom, such as the MF surface code, only needs to be able to implement the CNOT, Hadamard, and S- and T- gates. [5] In order to implement these gates, a sequence of measurements is performed via unitary transformations on the quantum states of several logical qubits defined by the operators {Xi} and {Zi}. The operators transform such that for unitary matrices U and W, UXiU = ±WXiW, where the sign is determined by the outcomes of specific measurements. The same relationship holds for Zi. These outcomes are interpreted by software and converted into the readout of a logical qubit. [4] The first gate to be constructed is the CNOT gate. From class, we know that the CNOT gate takes two qubits, a control and a target, and flips the value of the target qubit based on the value of the control, leaving the control value unchanged. For a two-qubit basis, the CNOT gate takes the matrix form: C = ( ) Such a gate can be constructed by braiding logical qubits in the MF surface code. The simplest example of a CNOT gate can be thought of as a single braiding operation that produces an overall sign change if the hexagonal ends of both qubits contain an anyon. In more complicated systems, ancilla qubits, extra qubits which have a secondary role in a logic circuit, are required to store outcomes of intermediate measurements. [4]

6 The second gate of interest is the Hadamard gate. Again, we know from class that the Hadamard gate exchanges the X and Z operators such that ĤX Ĥ = Z and ĤZ Ĥ = X where Ĥ has the matrix form: Ĥ = ( ( 1) ) For the standard bosonic surface code, the Hadamard operation requires a series of Hadamard gates acting on physical qubits enclosing the logical qubit. This results in an interchange between the X and Z stabilizers which is followed by multiple physical swap gates before completion. Luckily, the MF surface code greatly simplifies this process as the gate is realized by transferring a qubit between distinct sublattices so that the logical X and Z operators are exchanged. [4] This leaves only the S- T-gates to be examined. Figure 2: The circuit constructed from CNOT and Hadamard gates which result in the S-gate. [4] The S- and T-gate are phase-shift gates where the S-gate corresponds to a rotation around the z-axis by π/4 radians, and the T-gate, also known as the π/8 gate, is another z-axis rotation. [5] They can be written as the following matrices: S = ( i ) T = ( e iπ/4) Note that T 2 = S, S 2 = Z, S 4 = Î. Both of these gates can be constructed by performing a series of CNOT and Hadamard gates involving the logical qubit as well as an additional ancilla qubit. For the S-gate, the ancilla qubit must be in the state φs = ( +z + i -z ). Then, Hadamard and 1

7 CNOT gates will be applied to the system as shown in Fig. 2. For the T-gate, the ancilla qubit must be 1 in the state φt = ( +z + e iπ/4 -z ). [5] With the ancilla qubit ready, a probabilistic circuit can be designed in order to implement the T-gate as shown in Figure 3. First, a CNOT gate is performed between the ancilla and the logical qubit, Ψ. Next, the spin in the z-direction is measured, and an S-gate is applied. If Mz = 1, then the correct output is obtained. If Mz = -1, the result is a final state of ixzt Ψ. Therefore, the ix and Z operations must be undone by the surface code software before the correct output is obtained. [4] With the completion of these two gates, all four gates necessary for universal quantum computing have been addressed. Figure 3: The probabilistic circuit necessary to implement the T-gate. [4] As all four gates can be built using MF surface code, universal quantum computing is theoretically possible using systems with strong spin-orbit coupling. The qubits are protected from both flipping (classical) and phase shift (quantum) errors, and it is possible to construct hybrid systems that are resistant to noise. The main problems now can be traced to engineering. While feasible, constructing MF superconductors that can be used for universal quantum computing is experimentally out of reach. Hopefully as technology continues to advance, breakthroughs in spin-orbit coupling materials will result in a working MF universal supercomputer. References [1] C.W.J. Beenakker, Annu. Rev. Condens. Matter Phys. (4), (2013) [2] L.S. Ricco, et al., arxiv: v1 (2015). [3] Y. Li, arxiv: v1 (2015). [4] S. Vijay, et al., arxiv: v3 (2015). [5] A. G. Fowler, et al., Phys. Rev. A 86, (2012).