Fault-tolerant quantum computation is a basic problem in quantum computation, andteleportation is one of the main techniques in this theory. Using teleportation on stabi-lizer codes, the most well-known quantum codes, Pauli gates and Clifford operators canbe applied fault-tolerantly. Indeed, this technique can be generalized for an extended setof gates, the so called Ck hierarchy gates, introduced by Gottesman and Chuang (Nature,402, 390-392). Ck gates are a generalization of Clifford operators, but our knowledge ofthese sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev.A 77, 042313) raise the question of the relation between Ck hierarchy and the set ofsemi-Clifford and generalized semi-Clifford operators. They conjecture that any Ck gateis a generalized semi-Clifford operator. In this paper, we prove this conjecture for k = 3.Using the techniques that we develop, we obtain more insight on how to characterize Ckgates. Indeed, the more we understand Ck, the more intuition we have on Ck, k ≥ 4, and then we have a way of attacking the conjecture for larger k.
Fault-tolerant quantum computation is a basic problem in quantum computation, andteleportation is one of the main techniques in this theory. Using teleportation on stabi-lizer codes, the most well-known quantum codes, Pauli gates and Clifford operators canbe applied fault-tolerantly. Indeed, this technique can be generalized for an extended setof gates, the so called Ck hierarchy gates, introduced by Gottesman and Chuang (Nature,402, 390-392). Ck gates are a generalization of Clifford operators, but our knowledge ofthese sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev.A 77, 042313) raise the question of the relation between Ck hierarchy and the set ofsemi-Clifford and generalized semi-Clifford operators. They conjecture that any Ck gateis a generalized semi-Clifford operator. In this paper, we prove this conjecture for k = 3.Using the techniques that we develop, we obtain more insight on how to characterize Ckgates. Indeed, the more we understand Ck, the more intuition we have on Ck, k ≥ 4, and then we have a way of attacking the conjecture for larger k.
Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called Ck hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). Ck gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev. A 77, 042313) raise the question of the relation between Ck hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any Ck gate is a generalized semi-Clifford operator. In this paper, we prove this conjecture for k=3. Using the techniques that we develop, we obtain more insight on how to characterize C3 gates. Indeed, the more we understand C3, the more intuition we have on Ck, k≥4, and then we have a way of attacking the conjecture for larger k.
The Clifford hierarchy C1⊂C2⊂⋯ was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about Ck for k>3. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension d or with few qudits.
The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in ∪kCk is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case d=2, k=3 (arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of) U∈C3 on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group Γ1(U) generated by UPU† and beyond, we prove the conjecture for k≤4 and any prime dimension d.
Our proof centers on conjugation groups Γ1(U),Γ2(U),… of U∈Ck, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.
Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called Ck hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). Ck gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev. A 77, 042313) raise the question of the relation between Ck hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any Ck gate is a generalized semi-Clifford operator. In this paper, we prove this conjecture for k=3. Using the techniques that we develop, we obtain more insight on how to characterize C3 gates. Indeed, the more we understand C3, the more intuition we have on Ck, k≥4, and then we have a way of attacking the conjecture for larger k.
The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form C1ΠDC2 for Clifford gates C1,C2, a permutation gate Π, and a diagonal gate D; Beigi-Shor proved in 2008 that this holds for all third-level gates.
We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.
Teleportation is a crucial element in fault-tolerant quantum computation and a complete understanding of its capacity is very important for the practical implementation of optimal fault-tolerant architectures. It is known that stabilizer codes support a natural set of gates that can be more easily implemented by teleportation than any other gates. These gates belong to the so-called Ck hierarchy introduced by Gottesman and Chuang [Nature (London) 402, 390 (1999)]. Moreover, a subset of Ck gates, called semi-Clifford operations, can be implemented by an even simpler architecture than the traditional teleportation setup [X. Zhou, D. W. Leung, and I. L. Chuang, Phys. Rev. A 62, 052316 (2000)]. However, the precise set of gates in Ck remains unknown, even for a fixed number of qubits n, which prevents us from knowing exactly what teleportation is capable of. In this paper we study the structure of Ck in terms of semi-Clifford operations, which send by conjugation at least one maximal Abelian subgroup of the n-qubit Pauli group into another one. We show that for n=1,2, all the Ck gates are semi-Clifford, which is also true for n=3,k=3. However, this is no longer true for n>2,k>3. To measure the capability of this teleportation primitive, we introduce a quantity called ``teleportation depth,'' which characterizes how many teleportation steps are necessary, on average, to implement a given gate. We calculate upper bounds for teleportation depth by decomposing gates into both semi-Clifford Ck gates and those Ck gates beyond semi-Clifford operations, and compare their efficiency.
We study semi-Clifford and generalized semi-Clifford structures in the Clifford hierarchy as a function of hierarchy level k and qubit number n. We show that Ck(n) is semi-Clifford if and only if k≤2, n≤2, or k=3 and n≤6. We also establish that Ck(n) is generalized semi-Clifford when k≤4 or n≤3 and construct counterexamples when k,n≥5. For n=4 and k≥5, we show that every gate defined over Q(i) up to a scalar is generalized semi-Clifford, reduce the general case to a single level, and propose a strategy to prove our conjecture that Ck(n) is generalized semi-Clifford if and only if k≤4 or n≤4. Lastly, we go beyond this structure by introducing nc-generalized semi-Clifford gates, which we show are exactly the gates of the form ∑x∣π(x)⟩⟨x∣⊗Vx up to Clifford gates on both sides, where π permutes the computational basis states of nc control qubits. We conjecture that for k≥3 every gate in Ck(n) has this form for some nc≥1 with every Vx∈Ck−3(n−nc), refining the conjecture of Zeng, Chen, and Chuang.