Encoded Quantum Universality

Let us now discuss how to construct a universal set of 1-Ga gadgets satisfying properties 2. This construction is guided by the Ck gate hierachy in definition 2. We have already discussed that the Clifford group, C2, is notdense in SU(2n_{). In fact, the Gottesman-Knill theorem [90] shows that} quantum computation using qubits initialized in eigenstates of Pauli operators, Clifford-group gates, and measurements of Pauli operators can be simulated efficiently with a classical computer—let us call this set of operationsstabilizer operations or Gstab. But, by adding any onenon-Clifford gate to Gstab is sufficient to give a new set of operations which is dense in SU(2n) [33] and, hence, is quantum universal; examples of such gate sets were given in proposition 1.

5.2.2.1 Encoded Clifford-Group Operations

Although 1-Ga gadget constructions are known that apply to any stabilizer code [89], of practical interest are codes for which these constructions are efficient and give high accuracy thresholds. In fact, for some codes the construction of 1-Gas for the gates{H, S,cnot}that generate the Clifford group is particularly simple. Let us call a 1-Gatransversal if it can be realized by 0-Gas that act bitwise on the qubits in the code block (for a 1-Ga simulating a single-qubit gate), or bitwise between qubits at corresponding positions across different code blocks (for a 1-Ga simulating a multi-qubit gate). By construction, it is clear that transversal 1-Gas satisfy properties 2.

CSS codes provide a family of codes with such useful transversality properties [51, 89]: For any CSS code with k = 1 logical qubit, the logical_cnot can be implemented by bitwise_cnot gates. Moreover, if the CSS code is constructed from a self-dual classical code (i.e., if the quantum check matrix in equation (1.32) can be written so that H =H0), then the logical Hadamard gate can be implemented by bitwise Hadamard gates. If, in addition, this self-dual classical code is doubly even

(i.e., if all its code words have Hamming weight multiple of 4), then the logicalS gate is transversal so that the logical Clifford group can be generated by 1-Gas which are transversal. An example of a code with these properties is Steane’s [[7,1,3]] code [37] discussed in§1.3.4.

5.2.2.2 Quantum Software and Encoded Non-Clifford Gates

To complete our construction of 1-Gas satisfying properties 2 for a universal set of gates, it remains to construct a 1-Ga for a non-Clifford gate. This construction can be based on preparing a suitable ancillary state and then using a gate teleportation circuit [51, 42, 32, 77].

Consider the circuit shown in figure 5.5: To realize the single-qubit rotation by angleθ around the z-axis, Uz(θ) = exp −iθ

2Z

, we first prepare an ancillary qubit in the state |Aθi ≡ Uz(θ)|+i and, then, we perform a_cnotgate with the data qubit as control and the ancillary qubit as target. Finally, we measure the ancillary qubit in the computation basis. If the +1 eigenvalue is obtained, then we have successfully teleported the gateUz(θ); otherwise, when the outcome is−1,Uz(−θ) has been teleported instead and the correction operatorUz(2θ) needs to be applied.

|ψi • Uz(2θ) |Aθi

>= Z

Figure 5.5. A circuit that teleports the gate Uz(θ). An ancillary qubit is prepared in the state |Aθi ≡ Uz(θ)|+i, and then a cnot gate is executed as shown. Finally, the ancillary qubit is measured in the computation basis and the correction gate Uz(2θ) is applied if the measurement outcome is−1.

We observe that forθ=π/4, the teleported gate isT ≡Uz π₄

6∈C2. Furthermore, in this case the correction gate isS ≡Uz π₂

∈C2, and, so, a 1-Ga for simulating it can easily be constructed. It remains to construct a procedure for preparing thelogical|Aπ/4istate such that the 1-Ga for the T gate satisfies properties 2. Since the logical|Aπ/4iis just an ancillary state, its preparation can be done separate to the main computation; we can then view this quantum state as a form of “software” that we prepare and consume in order to implement fault-tolerant quantum computation.

The logical |Aπ/4i state is the +1 eigenstate of the logical Clifford-group operator T XT† = T2X = SX. Since we can construct a 1-Ga for the gate SX, we can also measure this operator using cat states: That is, we can measure the logicalSX by preparing a cat state, controlling each gate in the 1-Ga for SX from a different cat-state qubit and, finally, measuring all qubits of the cat state along the eigenbasis ofX and computing the parity of the measurement outcomes. Since a single fault in this measurement can cause an error in the eigenvalue of the logical SX operator that is measured, the measurement needs to be repeated sufficiently many times using different cat states with error-correction steps inserted between every two successive measurements.

the non-Clifford gateT that satisfies properties 2. An alternative construction for the non-Clifford Toffoli gate will be given in §5.3.4. Together with the 1-Ga constructions for Clifford-group gates discussed in the previous section, we have constructed 1-Gas satisfying properties 2 for a quantum universal set of gates.