The Clifford group is the set of gates generated by the controlled not gates, the Hadamard gate and the P={{1,0},{0,i}} gate. We will say that a n-qubit state is a Clifford state if it can be prepared using Clifford gates. In this paper we study the set of all 4-qubit Clifford states. We prove that there are 293760 states and their entanglement entropy must be either 0, 2/3, 1, 4/3 and 5/3. We also show that any pair of these states can be connected using local gates and at most 3 CNOT gates. We achieve this by splitting the 293760 states into 18 groups where each pair of states in a group can be connected with a local Clifford gate. We then study how the different CNOT gates act on the 18 groups. We also study the Clifford states with real entries under the action of the subgroup C_R of Clifford gates with real entries. This time we show that every pair of Clifford states with real entries can be connected with at most 5 CNOT gates and local gates in C_R. The link https://youtu.be/42MI6ks2_eU leads to a YouTube video that explains the most important results in this paper.
Let us denote by [Formula: see text] the Clifford group (the circuit or operations generated by Hadamard, [Formula: see text] phase and the controlled-NOT gates) and by [Formula: see text] the set of qubit states that can be prepared by circuits from the Clifford group. In other words, a state [Formula: see text] if [Formula: see text] where [Formula: see text]. We will refer to states in [Formula: see text] as Clifford states. This paper studies the set of all three-qubit Clifford states. We prove that [Formula: see text] has 8640 states and if we define two states [Formula: see text] and [Formula: see text] in [Formula: see text] to be equivalent if [Formula: see text], with [Formula: see text] a local transformation in [Formula: see text], then the resulting quotient space has five orbits. More exactly, [Formula: see text] where the orbit [Formula: see text] is made up of states with entanglement entropy [Formula: see text]. For example, the first orbit [Formula: see text] contains the state [Formula: see text] and corresponds to the unentangled Clifford states. We say that [Formula: see text] is a real state if all its amplitudes [Formula: see text] are real numbers. We also say that an operator is real if all the entries of its matrix representation with respect to the computational basis are real numbers. In this paper, we also study the set of real Clifford 3 qubits and the way this set splits when we identify two real Clifford states [Formula: see text] and [Formula: see text] to be equivalent if [Formula: see text] where [Formula: see text] is a local real Clifford operator. An interesting aspect that follows from this study of Clifford states is the existence of two real Clifford states [Formula: see text] and [Formula: see text] that can be connected with a Clifford local transformation but they cannot be connected with a real Clifford local transformation. This is, the equation [Formula: see text] for [Formula: see text], does have a solution in the set of local transformations from [Formula: see text] but it does not have a solution among the local transformations from [Formula: see text] that are real. We go a little deeper and show that the equation [Formula: see text] does not have a solution for any local operation (not necessarily Clifford) whose entries are real numbers. Finally, we show how the CNOT gates act on the set of Clifford states and also in the set of real Clifford states.
The Clifford group is the set of gates generated by controlled-Z gates, the phase gate and the Hadamard gate. We will say that a n-qubit state is a Clifford state if it can be prepared using Clifford gates. These states are known as the stabilizer states and they arise in quantum error correction. In this paper we study the set of all 5-qubit Clifford states. By using an exhaustive method we start by confirming that there are 19388160 states. The main goal of the paper is to understand the action of the controlled-Z gates action on the 5-qubit states. With this goal in mind, we partition the Clifford states into orbits using the equivalence relation: two states are equivalent if they differ by a local Clifford gate. We show that there are 93 orbits, and we label each orbit in such a way that it is easy to see the effect of the controlled-Z gates. Diagrams and tables explaining the action of the CZ gates on all the orbits are presented in the paper. A similar work is done for the real Clifford 5-qubits states, this is, for states that can be prepared with Controlled-Z gates, the Z gate and the Hadamard gate.
The Clifford group is the set of gates generated by the CZ gate, and the two local gates: the Hadamard and the Pi/2 phase shift gate. It is known that, for a two qubit system, the Clifford group C2 is a subgroup of order 92160 of the group of 4 by 4 unitary matrices. It is also known that the local Clifford gates LC2 is a subgroup of order 4608 of the group C2. In order to better understand the set C2, we make two matrices U1 and U2 in C2 equivalent if U_1=VU_2 for some V in LC2. We show that this equivalence relation splits C2 into 20 orbits, O1, ..., O20, each with 4608 elements. Moreover, for each orbit Oi, CZOi intersects 9 different orbits Oi1, ...,Oi9. Moreover, the intersection of Oij and CZOi has 512 matrices for each j=1,2, ..., ,9. The link https://www.youtube.com/watch?v=lcYtB2tnXFw&t=685s leads you to a YouTube video that explains the most important results in this paper.
It is well known that local gates have smaller error than non-local gates. For this reason it is natural to make two states equivalent if they differ by a local gate. Since two states that differ by a local gate have the same entanglement entropy, then the entanglement entropy defines a function in the quotient space. In this paper we study this equivalence relation on (i) the set
RQ(3) of 3-qubit states with real amplitudes, (ii) the set
QC of 3-qubit states that can be prepared with gates on the Clifford group, and (iii) the set
QRC of 3-qubit states in
QC with real amplitudes. We show that the set
QC has 8460 states and the quotient space has 5 elements. We have
∼QC={S0,S2/3,1,S2/3,2,S2/3,3,S1}. As usual, we will call the elements in the quotient space, orbits. We have that the orbit
S0 contains all the states that differ by a local gate with the state
∣000⟩. There are 1728 states in
S0 and as expected, they have zero entanglement entropy. All the states in the orbits
S2/3,1,S2/3,2,S2/3,3 have entanglement entropy
2/3 and each one of these orbits has 1152 states. Finally, the orbit
S1 has 3456 elements and all its states have maximum entanglement entropy equal to one. We also study how the controlled not gates
CNOT(1,2) and
CNOT(2,3) act on these orbits. For example, we show that when we apply a
CNOT(1,2) to all the states in
S0, then 960 states go back to the same orbit
S0 and 768 states go to the orbit
S2/3,1. Similar results are obtained for
RQC. We also show that the entanglement entropy function reaches its maximum value 1 in more than one point when acting on $\frac{\hbox{$\mathbb{R}$Q(3)} }{\sim}$.
We show that the problem of synthesizing a quantum circuit for a given Clifford operator using at most a specified amount of resources is NP-complete for several resource models of practical interest.
Specifically, we study synthesis of a given Clifford operator using single-qubit Clifford gates and two-qubit Clifford gates, with full qubit connectivity and no additional ancilla qubits. With single-qubit gates free and CNOT as the only two-qubit gate, we show that deciding whether a given target admits an implementation using at most a specified number of CNOT gates is NP-complete, and we show that the problem remains NP-complete when any two-qubit Clifford gate is allowed and the budget counts all such gates. Moreover, we show that both results continue to hold when SWAP gates may be inserted for free, and when single-qubit gates are also considered part of the budget.
In each case, we build on recent work by Li and Guan and show that there are polynomial-time reductions from Hamiltonian path in grid graphs to these decision problems, using a number of qubits quadratic in the number of grid vertices.