Clifford hierarchy demo

demonstrations_v2/tutorial_universality_and_clifford_hierarchy/demo.py

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+r"""Achieving universality with the Clifford hierarchy
+==========================
+
+
+We all know the 'Gottesman-Knill' rule: Clifford circuits are efficient to simulate but cannot provide quantum advantage on their own. We also know we need non-Clifford gates (like the $T$ gate) to reach universality. But why the $T$ gate specifically? Why not a random rotation?
+
+It turns out there is a rigorous structure hidden beneath these gates. The Clifford Hierarchy explains exactly how 'quantum' a gate is, how hard it is to correct, and why specific gates act as the 'magic' fuel for fault-tolerant computation.
+
+In this demo we will dig deeper into the levels of gates that make up the clifford hierarchy (the Pauli group, the Clifford group, T gates, and more), see how they are related (and how to implement them with gate teleportation!), and what this all means for FTQC.
+
+
+
+Achieving universal fault-tolerant quantum computing requires the ability to implement both Clifford and non-Clifford gates. Unfortunately, the Easton-Knill theorem proves that no quantum error correction scheme, necessary for protecting quantum data from noise, allows both Clifford and non-Clifford gates to be implemented transversally.
+
+
+The Easton-Knill theorem appears to make universality impossible in fault-tolerant quantum computing. However, the 
+
+
+
+What is the problem? 
+
+
+
+It appears like achieving the simultaneous goals of (a) universal (b) fault-tolerant quantum computing with the (c) potential for quantum advantage 
+
+
+On the surface, quantum computing that is simultaneously (a) universal, (b) fault-tolerant, and (c) potentially advantageous over classical methods appears extremely challenging because achieving one goals makes achieving the other harder. We want to protect 
+
+You may have heard statements like:
+* You don't ever need to physically execute Pauli correction gates
+* It's not possible to implement both Clifford and non-Clifford gates
+* You must have magic states for FTQC
+* It is possible 
+
+
+What is the answer?
+
+Executing a quantum program that can represent any unitary operation to acceptable precision, a property called universality, 
+
+## Definition of the Clifford hierarchy
+
+The Clifford hierarchy consists of infinitely nested sets of gates indexed as $k = 1, 2, \dots$ [Gottesman & Chuang]. To set things up, let's define the two lowest levels of the Clifford hierarchy. 
+
+### The Pauli Group ($C_1$)
+
+Members of the Pauli group, $C_1$, include the familiar Pauli gates: $\{X, Y, Z\}$. 
+
+### The Clifford Group ($C_2$)
+
+Members of the Clifford group, $C_2$, conjugate Pauli gates to Pauli gates, up to a global phase. 
+
+Formally, $C_2 = \{ U: U P U^\dagger \in C_1 \forall P \in C_1 \}$
+
+Members of this group include the Hadamard gate $H$, phase gate $S$, and the CNOT gate. As an example, they conjugate Paulis like so: $HZH^\dagger = X$ and $SXS^\dagger = Y$. Notice that the entire Pauli group lives within the Clifford group (e.g., $XZX^\dagger = -Z$), but the vernacular is that the Clifford group excludes the Pauli group i.e., $C_2 \setminus C_1$. The Gottesman-Knill theorem states that a circuit composed to entirely Pauli and Clifford gates are efficiently simulateable classically, meaning such circuits do not warrant making a quantum computer to execute. In this manner, these gates aren't too 'quantum'. 
+
+### $C_3$ set 
+
+At this stage, it is important to mention that many quantum error correction codes, such as the CSS, colour, surface, and qLDPC codes, have fault-tolerant and transversal implementations of gates belonging in $C_1$ and $C_2$ groups, making these gates straightforward to implement in a FTQC context. 
+
+However,
+
+1). The Gottesman-Knill theorem states that you must have a gate outside of $C_1$ and $C_2$ to have the chance of quantum advantage.
+2). The Eastin-Knill theorem states no quantum error correcting code can implement both Clifford and non-Clifford gates transversally.
+3). The proofs by Nebe, Rains, and Sloane (http://arxiv.org/abs/math/0001038v2) / The Solovay-Kitaev theorem show that you must have both Clifford and non-Clifford gates in your gate set to universally perform quantum computing. 
+
+In principle, you could select any gate that is outside of the Pauli and Clifford groups. Arbitrary rotation gates, for example, such as $R_Z(\theta)$ when $\theta \neq \{0, \pi/2, \pi\}$. 
+
+However, we shall see how members of the next level in the hierarchy, $C_3$, can efficiently address all three concerns. Members of $C_3$ are defined to satisfy: 
+
+$C_3 = \{U: U P U^{\dagger} \in C_{2} \forall P \in C_1\}$
+
+In other words, members of $C_3$ map to members of $C_2$ under conjugation, as in the Heisenberg picture. Examples of members of this group include the $T$ gate, Toffoli gate, and CCZ gate.
+
+### C_k set
+
+More generally, the $k^{\text{th}}$ level of the Clifford hierarchy for $k\geq 2$ is: 
+
+$C_k = \{U: UPU^{\dagger} in C_{k-1} \forall P \in C_1 \}$. 
+
+The Pauli and Clifford group constitute the foundation of an infinitely nested gates. 
+
+
+## Sidestepping the Eastin-Knill theorem -- Gate teleportation!
+
+Recall that many QEC codes naturally implement Clifford gates transversally and fault-tolerantly, but not non-Clifford gates such as $T$ gates. That's a consequence of the Eastin-Knill theorem. A non-transversal execution of a $T$ gate risks introducing irrecoverable noise to the encoded data. Transversal means that each qubit of a logical qubit interacts with itself or its counterpart in another logical qubit to prevent the spread of errors. Therefore, we need a method to safely apply a non-Clifford gate to logical qubits. 
+
+Gottesman and Chuang showed that it is always possible to apply a gate in $C_k$ with a gate teleportation circuit using gates and measurements that are at most in $C_{k-1}$. 
+
+Gate teleportation is an extension of state teleportation (recall Alice and Bob). To teleport a $T$ gate, see Figure 1 below. Prepared in advance is a Bell state aka EPR pair where the $T$ gate is applied to half of the pair. The other half undergoes Bell basis measurement with the input state $| \psi \rangle$. Such measurements have a uniform probability of introducing a Pauli $\union ~I$ gate change. 
+
+Knowing the definition of a $C_3$ gate leads to $T P T^{\dagger} = C$, where $C$ is some Clifford gate that can be classically determined once the Bell basis measurement reveals $P$. Hence, the state may be written as $TP|\psi\rangle = CT| \psi \rangle$. If we apply $C^{\dagger}$, conditioned on the result of $P$, then the output state becomes $T| \psi \rangle$. 
+
+Fault tolerance becomes a question of careful state preparation rather than 
+
+
+
+## Teleportation is more efficient with semi-Clifford gates, like $T$ gates
+
+## Interesting mathematical properties & consequences of the Clifford hierarchy
+
+Note that higher level diagonal gates are interesting to be able to implement too. The multi-controlled Z gates that appear in Shor's period finding algorithm or in QFT, are one example. Here, the $C^kZ$ gate resides in the $k+1$ level of the Clifford hierarchy. 
+
+
+
+"""

demonstrations_v2/tutorial_universality_and_clifford_hierarchy/metadata.json

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+            "authors": "Sergei Bravyi and Alexei Kitaev",
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