Implementing arbitrary phase gates with Ising anyons
Abstract
Ising-type non-Abelian anyons are likely to occur in a number of physical systems, including quantum Hall systems, where recent experiments support their existence. In general, non-Abelian anyons may be utilized to provide a topologically error-protected medium for quantum information processing. However, the topologically protected operations that may be obtained by braiding and measuring topological charge of Ising anyons are precisely the Clifford gates, which are not computationally universal. The Clifford gate set can be made universal by supplementing it with single-qubit -phase gates. We propose a method of implementing arbitrary single-qubit phase gates for Ising anyons by running a current of anyons with interfering paths around computational anyons. While the resulting phase gates are not topologically protected, they can be combined with “magic state distillation” to provide error-corrected -phase gates with a remarkably high threshold.
pacs:
03.67.Lx, 03.65.Vf, 03.67.Pp, 05.30.PrNon-Abelian anyons – quasiparticles with exotic exchange statistics described by multidimensional representations of the braid group Leinaas and Myrheim (1977); Goldin et al. (1985); Fredenhagen et al. (1989); Fröhlich and Gabbiani (1990) – can provide naturally fault-tolerant platforms for quantum computation. The non-local state space of such anyons can be used to encode qubits that are impervious to local perturbations. Topologically protected computational gates may be implemented by braiding the anyons Kitaev (2003); Preskill (1998); Freedman (1998); Freedman et al. (2002) or by measuring their topological charge Bonderson et al. (2008, 2009).
Ising-type anyons
The braiding transformations of Ising anyons are given by the spinor representations of SO Nayak and Wilczek (1996). The set of gates that may be obtained through braiding and/or topological charge measurement of Ising anyons is encoding-dependent, but never computationally universal. For the standard qubit encoding (i.e. one qubit in four anyons), the computational gates obtained via braiding or measurement of anyon pairs are the single-qubit Clifford gates. These gates can be generated by the Hadamard and -phase gates,
(1) |
where is called the “-phase gate.”
The controlled-NOT gate
(2) |
may be implemented by allowing the use of non-demolitional measurements of the collective topological charge of four anyons Bravyi and Kitaev (2005); Bravyi (2006); Bonderson (). Adding this generates the full set of Clifford gates, which can be efficiently simulated on a classical computer, but becomes universal when supplemented with a single-qubit -phase gate Boykin et al. (1999).
One way to obtain -phase gates (as well as CNOT gates) is through dynamical topology change of the system Bravyi and Kitaev (2000); Freedman et al. (2006). However, this requires complicated physical manipulations of the system which are (at best) currently infeasible, such as switching between planar and non-planar geometries.
Alternatively, if one can implement ideal (e.g. topologically protected) Clifford gates, then they can be used to perform “magic state distillation” Bravyi and Kitaev (2005); Bravyi (2006) to produce error-corrected -phase gates from noisy ones. This purification protocol (which has poly-log overhead) consumes several copies of a magic state, e.g. , and outputs a single qubit with higher polarization along a magic direction. Once a sufficiently pure magic state is produced, it may then be consumed to generate a -phase gate. This protocol permits a remarkably high error threshold of over for the noisy gates, as compared to the “high” threshold of for postselected quantum computation Aliferis et al. (2008). Hence, it is important to devise practical methods of generating the -phase gate within this error threshold for systems with Ising anyons.
A simple proposal for this is to move bulk quasiparticles close enough to each other to let the microscopic physics split the energy degeneracy of the fusion channels encoding a qubit. The resulting time evolution can produce arbitrary phase gates, albeit unprotected ones in need of error-correction (e.g. by magic state distillation). However, the energy splitting caused by bringing two quasiparticles together oscillates rapidly with their separation Baraban et al. (2009); Cheng et al. (2009), so small errors in the quasiparticles’ spatial separation will translate into large errors in the phase. Thus, this approach appears unlikely to be able to meet even the generous error threshold of magic state distillation.
In this letter, we propose a method of implementing arbitrary phase gates for systems with Ising-type anyons that aims to be more practical and to achieve a manageable error rate. This method involves a device consisting effectively of a beam-splitter or tunneling junction that is used to run a current of anyons through interfering paths around computational anyons. We first analyze the effect of such a device using a semiclassical picture of the anyonic current applicable to general Ising systems. Subsequently, we perform a more detailed analysis (including error estimates) for Ising-type systems in which the anyonic current is provided by edge modes described by conformal field theory.
For the purpose of constructing the phase gate, we consider a topological
qubit encoded in a pair of anyons carrying Ising topological charge
In other possible physical realizations of Ising anyons, the anyons comprising a qubit may need to be pinned by other means, e.g. in a chiral -wave superconductor, a hole may be bored through the sample where flux can be trapped. It may be also be easier in some realizations to construct interfering paths for a beam of bulk quasiparticles, rather than to rely on edge quasiparticles. With this situation in mind, we now compute semiclassically the effect of a beam of quasiparticles incident from the left. This calculation will also capture some of the features of the more involved edge theory calculation, relevant to the quantum Hall setting. For ease of comparison with Fig. 1, we use terminology appropriate to that picture. We assume that at the tunneling junction in Fig. 1 a quasiparticle can tunnel with amplitude and will continue along the edge with amplitude .
We can treat the motion of the anyons semiclassically and analyze the effect that sending them through the device has on the qubit. This effect results from the braiding statistics, which contributes a factor of or when a anyon travels one full circuit around a region containing topological charge or , respectively. This non-Abelian contribution is in addition to the Abelian phase acquired when a anyon travels once around the device loop counterclockwise. This phase contains the Abelian statistical angle, the Aharanov-Bohm phase, and possibly other terms, depending on the specific realization of the device.
The resulting transformation to the qubit when one anyon has passed through the device is
(3) | |||||
where accounts for the non-Abelian braiding statistics. Here the first term results from direct tunneling across the constriction, and the remaining terms describe the effect of the quasiparticle passing around the edge of the sack one or many times. This does not transfer topological charge to the qubit, so the matrix is diagonal and unitary. However, braiding a quasiparticle from the beam around the computational anyons is topologically equivalent to processes that transfer topological charge between the computational pair. These are the same processes that would cause energy splitting between the otherwise degenerate fusion channels of the qubit when its quasiparticles are brought close together Bonderson (2009). Hence, the net effect of passing a through the device is similar to that of splitting the energy, i.e. to produce a relative phase between these channels. Up to an overall phase, where
(4) |
and . For , this gives . The phase gate generated using this device may be controlled by sending multiple quasiparticles through the system, or by adjusting the experimental variables and .
For Ising-type systems that support an anyonic edge current, such as those in Refs. Moore and Read (1991); Blok and Wen (1992); Lee et al. (2007); Levin et al. (2007); Bonderson and Slingerland (2008); Read and Green (2000); Kitaev (2006); Fu and Kane (2008); Sau et al. (2009), we should go beyond this semiclassical calculation and analyze the quasiparticle tunneling and interference using the proper edge theory. The combined edge and qubit system is described by the Hamiltonian
(5) |
where is the Hamiltonian describing the unperturbed edge and describes tunneling of quasiparticles across the constriction. As before, the represents the braiding statistics of the edge with the qubit, picking up a minus sign each time the braids around the charge. The strength of the tunneling Hamiltonian can be adjusted by changing the separation distance across the sack constriction. We represent the density matrix of the combined system by and the qubit’s density matrix is obtained from this by tracing out the edge .
Solving the interaction picture Schrödinger equation
(6) |
where , with the assumption that the edge and qubit are unentangled at time , we find that
(7) |
where and are real-valued time dependent quantities, and . The diagonal elements of the qubit density matrix are unaltered from their initial state, as the Hamiltonian commutes with . The sack geometry will therefore implement a phase gate with phase-damping noise parameterized by .
Applying a Hadamard gate and then this noisy phase gate to an initial state creates a magic state with error
(8) |
If , then this can be used with magic state distillation to generate an error-corrected -phase gate Bravyi and Kitaev (2005); Bravyi (2006).
Computing the values of and to second order in the tunneling Hamiltonian, we have
(9) | |||||
(10) |
We note that takes the form of a variance in the phase.
To compute concrete values of and for the most physically relevant example, we turn to the field theoretic description of the edge of a Moore-Read (MR) quantum Hall state Moore and Read (1991). The Lagrangian for the unperturbed edge is Fendley et al. (2007)
(11) |
where the charged and neutral sectors are respectively described by the chiral boson () and fermion () modes, with velocities and . The operator that tunnels quasiparticles with charge across the constriction is
(12) |
where includes the Aharanov-Bohm phase () acquired in traveling around the sack as well as any Abelian braiding statistics factors.
Assuming the edge was initially in thermal equilibrium at temperature , i.e. , we find
(13) |
Here is a short range cutoff, and are the scaling exponents of the charge and neutral modes, respectively, and .
From Eqs. (9,13), we see that there are several experimental parameters which may be used to control the phase generated using the sack geometry. In particular, we envision and the area enclosed in the sack as the primary physical quantities to adjust, since these provide a practical means of tuning and , respectively, while keeping the other quantities essentially constant. With a properly designed geometry, these quantities can be adjusted sufficiently while causing only negligible changes to . In contrast to the tunneling amplitude of neutral excitations, which oscillates rapidly with distance Baraban et al. (2009); Cheng et al. (2009) (and can be understood as Friedel oscillations in a composite fermion picture), the tunneling amplitude of quasiparticles does not oscillate and decays as for , where is the magnetic length Bishara et al. (2009); Chen et al. (2009).
There are several ways to adjust the phase for quantum Hall systems. One practical method is to alter the total area enclosed in the sack by using a side gate. This leads to a change in the flux enclosed in the two interfering current paths, and thus a change in the Aharanov-Bohm phase included in . Another method for changing is by applying a current along the edge of the system. This may be implemented via a voltage difference between the edge that forms the sack structure and the edge on the other side of the electron gas. Driving this current populates or depopulates charge on the edge of the electron gas, and hence changes the area as a side gate would.
Let us hold fixed all the experimental parameters except the tunneling amplitude, which we vary as , for a general (real, non-negative) signal profile with characteristic “duration” time scale . This gives , where
(14) | |||||
(15) |
and
(16) | |||||
(17) | |||||
(18) |
where . We note that exponentially for long times with a decay rate proportional to the temperature. We generally have the bound
(19) |
where is a dimensionless function of , , and . When (i.e. is much shorter than the thermal coherence length), this becomes a temperature independent bound with now depending only on , , and . Using in these expressions, we see that it is favorable to increase the duration (e.g. by using weaker tunneling) used to enact a particular phase gate, since the bound decreases as . However, one must obviously balance this with the need to keep time scales much shorter than the qubits’ coherence time.
To demonstrate that the -phase gate can (at least in principle) be implemented with sufficiently low error using this device, we compute resulting for using a sack of length m, a rectangular pulse of duration , i.e.
(20) |
(21) |
and velocities m/s and m/s estimated for the state from numerical studies Wan et al. (2008); Hu et al. (2009). In Fig. 2, we display the resulting region of parameter space (for different values of ) in which the error is below the threshold for magic state distillation. The threshold curves move up as is varied away from , and will diverge as or . However, this divergence is evidently not problematic unless is rather close to the singular points.
It is straightforward to repeat the preceding edge theory analysis for other Ising-type systems with a conformal edge theory. The results are again given by the preceding equations, but with different values of , , , and . The values of these quantities for Ising-type quantum Hall candidates for all the observed second Landau level plateaus, such as the MR state Moore and Read (1991), SU NAF state Blok and Wen (1992), the anti-Pfaffian state Lee et al. (2007); Levin et al. (2007), and the Bonderson-Slingerland hierarchy states Bonderson and Slingerland (2008) built on any of these Ising-type states, can be found in Bishara et al. (2009). For systems with chargeless Ising edges Read and Green (2000); Kitaev (2006); Fu and Kane (2008); Sau et al. (2009), one has and .
We also note that we can use our device to generate the two-qubit gate: by putting two pairs of anyons, each pair corresponding to a separate encoded qubit, into the sack. This is an entangling gate if for , and in particular is a Clifford gate when . Similarly, this device can be used to generate multi-qubit gates.
In addition to offering a correctable error rate, the phase gate implementation described herein offers several advantages that increase its practicality. This device may be utilized in a manner compatible with proposals for “measurement-only” topological quantum computation Bonderson et al. (2008, 2009). Specifically, the computational anyons may remain stationary while only the edge of the system is manipulated, thus circumventing the need for fine control over the motion of bulk quasiparticles. As this device would only require the use of established techniques for deforming the edge using top and side gates Willett et al. (2009), it provides the first realistic proposal for achieving universal quantum computation using Ising anyons.
Acknowledgements.
We thank S. Simon for useful discussions. DC and KS acknowledge the support and hospitality of Microsoft Station Q. DC, CN and KS are supported in part by the DARPA-QuEST program. KS is supported in part by the NSF under grant DMR-0748925.Footnotes
- We use the term “Ising-type” to include anyons which only differ from Ising by Abelian factors, e.g. can be obtained from Ising through products or cosets with U sectors.
- Strictly speaking, more than two anyons are needed to comprise a topological qubit and we will assume the standard one qubit in four anyons encoding, but the operation we perform only requires manipulations involving two of these anyons.
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