The most prominent drawback of the technology is that qubits are inherently fragile. Hence, they can only retain quantum information for a tiny amount of time. Interactions within the system and with a noisy environment are typically the limiting factors of coherence time: the duration during which the underlying qubit system retains its quantum properties. Therefore, the best devices require extremely clean control signals and cryogenic operation to reduce thermal noise.
Those random fluctuations will occasionally flip or randomise the state of a qubit, potentially corrupting the computation. Hence, to create a “functional” quantum com- puter, we need a quantum computer that is fault-tolerant in the sense that the quantum computer must be able to detect failures so that they do not spread in space and time during computation and hence can be corrected.
One of the most prominent contributions of fault-tolerant quantum computing re- search is the invention and the continuous improvement of various quantum error correction (QEC) codes [52, 53, 30, 148, 32, 161, 189, 43, 42, 162, 167, 164, 118], which provide a potential pathway to achieving universal quantum computing.
12 Chapter 1. Introduction
The goal of quantum error correction is to use redundancy and correction to realise logical qubits with improved error rates as compared with that of the elementary qubits. The fault-tolerance threshold [99], which is the qubit failure probability below which reliable quantum computation becomes feasible, is the standard measure of fault-tolerant quantum computing, where error propagation is constrained such that error correction protocols remain effective [54].
Among a wide variety of quantum error correction codes, thesurface code[30, 94, 45, 189, 43] has stood out in terms of computational error threshold which is about two orders of magnitude higher than that of conventional concatenated coding schemes. The critical feature is that implementing the surface code requires a regular 2-D arrangement of qubits, where neighbouring qubits interact with each other in a pairwise manner and in parallel (see Fig. 1.5). Qubits are classified either asdataqubits orsyndrome(ancilla) qubits according to their roles in the quantum error correction procedure. Each syndrome qubit measurement fixes an eigensubspace of astabilizeroperator, which involves all four neighbouring data qubits. Logical qubits are defined as topologicaldefectson the qubit lattice where syndromes are not measured. Thus, there are two types of logical qubits, so-calledsmooth(Z-cut) andrough(X-cut) logical qubits. The codedistanceis defined either by the perimeter of the defects or the distance between them, whichever is smaller. Interested readers should consult Fowleret al.[43] for an in-depth review.
For surface code, there are only two types of syndromes: ZorX syndromes, which stand for ZZZZ or X X X X operators acting on the four data qubits. The Z and X
syndromes are measured by performing a sequence ofCNOT gates between the ancilla and its four neighbouring data qubits as shown in the quantum circuit diagrams in Fig. 1.6a and 1.6b, respectively.
Logical qubits are defined as topologicalholes(defects) on the qubit lattice where syndromes are not being measured. Thus, there are two types of logical qubits, so-called
smooth(Z-cut) and rough (X-cut) logical qubits. The code distance is then defined either by the perimeter of the holes or the distance between them, whichever is smaller. The reason is that an uncorrectable error occurs in surface code whenever physical qubit errors form a continuous chain surrounding a hole or connecting two holes. It is worthwhile mentioning that the boundary of the code lattice could also be considered
1.2 Quantum Computing 13
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Fig. 1.5 Surface code layout: white circles represent data qubits; filled circles are syndrome qubits (X stabilizers in green and Z stabilizers in yellow). Each internal stabilizer acts on four adjacent data qubits, while boundary stabilizers act on either two or three data qubits.
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(a)Z-syndrome measurement
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(b)X-syndrome measurementFig. 1.6 Quantum circuit to measure surface code syndromes. The first line represents the ancilla, while the following lines represent the surrounding data qubits.
as a superficial hole. Therefore, crossing the boundary from one end to the opposite or connecting one hole to the boundary are also irreversible logical errors. Interested readers should consult [43] for an in-depth explanation.
The bigger and farther apart the surface code holes are, the higher code distance we can get. Therefore, the ability to fabricate a gigantic lattice of qubits is pivotal in making the surface code to work. Besides, because the logicalCNOT operation in the
14 Chapter 1. Introduction
surface code involves braiding a pair of holes as shown in Fig. 1.7, the actual distance between two distance-dholes needs to be at least 3d. An example of theCNOT braiding evolution [43] between anX-cut qubit and aZ-cut counterpart is illustrated in Fig. 1.8. In the surface code layout, a pair of holes that are 3dapart will serve as a unit cell. Braiding operations can be optimised at the architectural level with regard to the computation time or lattice area as shown in [135]. At the micro-architectural level, on the other hand, we need to be able to keep track of the holes’ locations as well as grow and shrink them by turning off and on the respective syndrome qubits.
1st pair of holes
2nd pair of holes
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Fig. 1.7 Braiding diagram of a logicalCNOT gate in surface code with time axis running horizontally.
The code distance is calculated by the intrinsic error rate of the underlying qubit hardware as well as the threshold of the selected error correction code, as following [86]
εlogical≥C1 C2εphysical εthreshold ⌊d+1 2 ⌋ , (1.8)
whereC1andC2are constants that are code-dependent. For surface code, according to
[46], the values forC1andC2are 0.13 and 0.61, respectively. We denote the error rate per logical gate operation asεlogical, which depends on the target success probability of the algorithm and the number of logical gate operations to complete it. An entire algorithm success rate ofεalgowill requireεlogical ≈εalgo/Nlogical, whereNlogical is the number of logical gate operations required for the entire computation. The physical error rateεphysical is the experimental fidelity of the qubits, which is compared against the threshold of the surface code atεthreshold ≈1% to determine the minimum code distance
1.2 Quantum Computing 15
3d
(a) Original holes (b) Growth hole A
(c) Shrink hole A (d) Growth hole A
Fig. 1.8 LogicalCNOT gate by braiding holes in surface code. The braiding is done by expanding and shrinking a hole. In order to maintain the code distancedthroughout this process, the orginal holes need to be separated by at least 3d.
d. This is the minimum code size that can guarantee the algorithm to be completed with high probability.
From this code distance estimation, we can asymptotically approximate the number of physical qubits needed and therefore the expected classical resources. Remember that a distancedlogical qubit needs a hole covering approximatelyd2physical qubits. In addition, all the holes need to be 3d apart to make sure logicalCNOT gates can be readily performed between any pair of logical qubits. A simple calculation can be made
16 Chapter 1. Introduction
to estimate the number of physical qubits as follows.
Nphysical∝d2×(4×pNlogical)2, (1.9) in which, the number of logical qubitsNlogical is algorithm dependent, and the code distancedis from the previous estimation step.