The robustness of magic state distillation against errors in Clifford gates
Abstract
Quantum error correction and faulttolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for faulttolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state Bravyi. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than faulttoleranceassisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a faulttolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future smallscale quantum computing devices as faulttolerance need only be used in the final steps of the protocol.
pacs:
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I Introduction
Processes such as imperfect control of quantum operations or unintended coupling between qubit systems and their environment lead to errors in any realistic implementation of a quantum computing device. As such, quantum error correction has been developed in order to recover the quantum information that would otherwise be lost due to these faults Landauer; ShorEC; SteaneEC. However, quantum error correction itself is not enough for the implementation of a robust quantum computing device as errors that occur between the error correction steps could propagate between qubits. Propagating errors could prove to be detrimental to the recovery of quantum information and need to be avoided in order to implement any realistic error correction scheme. Faulttolerant quantum computation aims to address this concern by encoding the information of each qubit into a larger Hilbert space of many qubits and perform encoded quantum gates in such a way that errors do not propagate through multiple qubit blocks ShorFT. Faulttolerance allows the faults to remain tractable at the cost of needing a polylogarithmic increase in the number of qubits and quantum gates to perform encoded operations when the error rate of the quantum gates is below a certain target threshold Gottesman97; KLZ; Preskill; Aharonov; AGP06.
A desired property of encoded gate operations is transversality, which prevents the propagation of errors in the encoded states of the faulttolerant architecture. The Clifford gate set, the group of gates generated by the Hadamard gate, the phase gate, and CNOT gate, has been shown to be transversal for many quantum codes Nielsenbook. The Clifford gates, along with measurement and state preparation in the eigenbasis form the class of stabilizer operations, which have been shown to be efficiently classically simulatable AG04fromBen. In order to use Clifford gates for universal quantum computation, one requires the ability to produce a pure, singlequbit, nonstabilizer state Knill2004, known as a magic state. Perfect magic state preparation is difficult due to experimental errors, however magic state distillation allows for the creation of an arbitrarily high fidelity magic state from noisy ancillas by repeatedly applying stabilizer operations Bravyi; Ben; Ben2. In this work we investigate the effect of noise, in the state preparation and quantum gate application, on the convergence of the magic state distillation protocol.
In Bravyi and Kitaev’s magic state distillation protocol Bravyi, five copies of a noisy magic state are used to extract a single state of higher fidelity with respect to the magic state. The process is then iterated to increase the fidelity to be arbitrarily high. The input states to the distillation scheme are assumed to be along the magic state axis in the Bloch sphere, the axis connecting the magic state and its orthogonal complement. This assumption is based upon the ability to perform a dephasing channel that collapses all points of the Bloch sphere to their projection along this axis. In this work, we extend the analysis of the distillation scheme to one where the location of the input state is an arbitrary state in the Bloch sphere. The motivation for this is twofold, performing a dephasing operation in an experimental setup would introduce errors that would manifest themselves as perturbations off the magic state axis; moreover, there may be states off the magic state axis that converge faster to the magic state after multiple iterations than the states with the same fidelity on the magic state axis. Performing such an analysis will enable us to conclude that the magic state distillation protocol is robust to slight perturbations about the magic state axis in the Bloch sphere due to noise.
In Section III, we turn our attention to the effect of noise present in the quantum gates of the distillation protocol. We investigate the consequences of depolarizing noise, a oneparameter noise model, on the one and twoqubit Clifford gates in the fivequbit decoder of the protocol. Such noise will affect the rate of convergence, increase the threshold for the input states of the protocol, and pose a restriction on the ability to prepare a magic state with arbitrarily high fidelity. As such, with the development of faulttolerant schemes for the reduction of noise, one can ask if it is more efficient in the total number of quantum gates to use faulty gates or faulttolerant encoded gates to perform the distillation protocol. In Section LABEL:sec:GateComparison we show that using faulty Clifford gates is the more efficient scheme and using a faulttolerant encoding is unnecessary unless the required gate fidelity of the universal gate is much higher than that of the Clifford gates at one’s disposal. The ability to perform magic state distillation without the use of a faulttolerant encoding is promising for future experimental realizations of multiple rounds of state distillation on smallscale quantum computers.
Ii The evolution of quantum states under magic state distillation with perfect Clifford gates
The input states to Bravyi and Kitaev’s magic state distillation protocol are assumed to be along the magic state axis Bravyi. However, preparing such states may prove to be difficult experimentally. In this section we study the convergence of the distillation scheme under perturbations about the magic state axis. Moreover, we show that for low fidelity input states, the convergence to the magic state may be improved for states away from the magic state axis. This suggests that while performing a dephasing operation to initialize the input states to be along the axis may be useful, it is not absolutely necessary in certain fidelity regimes.
The Clifford gate set is generated by the following gates,
(1) 
where each individual gate can be performed on any qubit, and CNOT can be performed on any pair of qubits where and are 2–by–2 Pauli matricies. The power of magic state distillation is that it requires only the use of Clifford gate operations, along with state ancilla preperation and measurement in the basis, to distill multiple copies of noisy magic states to one of higher fidelity, provided the initial state is above a given threshold. This is appealing as Clifford gates have been shown to have transversality in many quantum error correcting codes Nielsenbook, and as such can be implemented faulttolerantly in order to reduce their error rate.
Let denote the Pauli matrix in the computational basis. There are two types of magic states, up to onequbit Clifford operators, the Htype magic state,
which can be used to implement the phase gate, , and the Ttype magic state,
which can be used to implement the phase gate Bravyi, both of which, along with the Clifford gates, provide universal quantum computation. Many efforts have contributed to building protocols for magic state distillation and achieving tight noise thresholds for the noisy input ancillas to the distillation protocol in order to understand the transition from classically simulatable quantum computation to genuine quantum computation Bravyi; Ben; Ben2; Campbell1; Campbell2; Anwar; Veitch; Anderson. Additionally, an experimental demonstration of a single round of Bravyi and Kitaev’s distillation protocol of Ttype magic states has been performed in Nuclear Magnetic Resonance (NMR) jingfu.
The first step of Bravyi and Kitaev’s distillation protocol Bravyi is to perform a dephasing operation on five copies of the initial state of the quantum system,
(2) 
where is a Clifford group gate. If the initial state of the system is expressed according to its Bloch sphere coordinates ,
(3) 
the transfomation is equivalent to projecting the state onto the magic state axis connecting the states and in the Bloch sphere, where is the state orthogonal to ,
(4) 
The dephasing operation leads to the ability to derive a clean threshold for the input fidelity of the initial states in order for the magic state distillation protocol to be beneficial. However, errors in the implementation of the quantum information processor could lead to a preparation of states away from the magic state axis. In this section, we provide an analysis of the effectiveness of the magic state distillation protocol for states prepared at an arbitrary location in the Bloch sphere and give modified target fidelities for state distillation under such noisy state preparation.
The distillation protocol consists of the above dephasing transformation on five prepared initial states, followed by a measurement of the stabilizers of the fivequbit code Bennett96; Laflamme96, , , , , and decoding upon obtaining the “+1” eigenstate of all the stabilizers Bravyi, where and . The roles of the measurement and decoding can be reversed, and the overall circuit can be described by the diagram in Figure LABEL:fig:perfect_circuit. Various encoding/decoding circuits can be developed to encode the five qubit code, we chose to analyze the circuit presented in Figure LABEL:fig:perfect_circuit as once a qubit is used as a control qubit in a twoqubit gate, it is no longer used in any further twoqubit gates, thus minimizing the propagation of errors through the circuit Grassl. Upon following the steps outlined in the above procedure, the initial noisy states must have a fidelity greater than Bravyi, where the fidelity with respect to the magic state is defined as . Repeating the protocol to obtain multiple copies of the state of increased fidelity , the process can be iterated to obtain magic states with arbitrarily high fidelity. with respect to the magic state in order for the output state to be of higher fidelity
