Conventions
States and Bases
Bases
A basis refers to a set of two eigenstates. The transition between these two states is said to be addressed by a channel that targets that basis. Namely:
Basis 
Eigenstates 



\(g\rangle,~r\rangle\) 


\(g\rangle,~h\rangle\) 


\(0\rangle,~1\rangle\) 

Qutrit state
The qutrit state combines the basis states of the groundrydberg
and digital
bases,
which share the same ground state, \(g\rangle\). This qutrit state comes into play
in the digital approach, where the qubit state is encoded in \(g\rangle\) and
\(h\rangle\) but then the Rydberg state \(r\rangle\) is accessed in multiqubit
gates.
The qutrit state’s basis vectors are defined as:
Qubit states
Warning
There is no implicit relationship between a state’s vector representation and its associated measurement value. To see the measurement value of a state for each measurement basis, see State Preparation and Measurement .
When using only the groundrydberg
or digital
basis, the qutrit state is not
needed and is thus reduced to a qubit state. This reduction is made simply by tracingout
the extra basis state, so we obtain
groundrydberg
: \(r\rangle = (1, 0)^T,~~g\rangle = (0, 1)^T\)digital
: \(g\rangle = (1, 0)^T,~~h\rangle = (0, 1)^T\)
On the other hand, the XY
basis uses an independent set of qubit states that are
labelled \(0\rangle\) and \(1\rangle\) and follow the standard convention:
XY
: \(0\rangle = (1, 0)^T,~~1\rangle = (0, 1)^T\)
Multipartite states
The combined quantum state of multiple atoms respects their order in the Register
.
For a register with ordered atoms (q0, q1, q2, ..., qn)
, the full quantum state will be
Note
The atoms may be labelled arbitrarily without any inherent order, it’s only the
order with which they are stored in the Register
(as returned by
Register.qubit_ids
) that matters .
State Preparation and Measurement
Basis 
Initial state 
Measurement 


\(g\rangle\) 
\(r\rangle \rightarrow 1\)
\(g\rangle,h\rangle \rightarrow 0\)


\(g\rangle\) 
\(h\rangle \rightarrow 1\)
\(g\rangle,r\rangle \rightarrow 0\)


\(0\rangle\) 
\(1\rangle \rightarrow 1\)
\(0\rangle \rightarrow 0\)

Measurement samples order
Measurement samples are returned as a sequence of 0s and 1s, in
the same order as the atoms in the Register
and in the multipartite state.
For example, a fourqutrit state \(q_0, q_1, q_2, q_3\rangle\) that’s projected onto \(g, r, h, r\rangle\) when measured will record a count to sample
0101
, if measured in thegroundrydberg
basis0010
, if measured in thedigital
basis
Hamiltonians
Independently of the mode of operation, the Hamiltonian describing the system can be written as
where \(H^D_i\) is the driving Hamiltonian for atom \(i\) and \(H^\text{int}_{ij}\) is the interaction Hamiltonian between atoms \(i\) and \(j\). Note that, if multiple basis are addressed, there will be a corresponding driving Hamiltonian for each transition.
Driving Hamiltonian
The driving Hamiltonian describes the coherent excitation of an individual atom between two energies levels, \(a\rangle\) and \(b\rangle\), with Rabi frequency \(\Omega(t)\), detuning \(\delta(t)\) and phase \(\phi(t)\).
Warning
In this form, the Hamiltonian is independent of the state vector representation of each basis state, but it still assumes that \(b\rangle\) has a higher energy than \(a\rangle\).
Pauli matrix form
A more conventional representation of the driving Hamiltonian uses Pauli operators instead of projectors. However, this form now depends on the state vector definition of \(a\rangle\) and \(b\rangle\).
Pulser’s statevector definition
In Pulser, we consistently define the state vectors according to their relative energy. In this way we have, for any given basis, that
Thus, the Pauli and excited state occupation operators are defined as
and the driving Hamiltonian takes the form
Alternative statevector definition
Outside of Pulser, the alternative definition for the basis state vectors might be taken:
This changes the operators and Hamiltonian definitions, as rewriten below with highlighted differences.
Note
A common case for the use of this alternative definition arises when
trying to reconcile the basis states of the groundrydberg
basis
(where \(r\rangle\) is the higher energy level) with the
computationalbasis statevector convention, thus ending up with
Interaction Hamiltonian
The interaction Hamiltonian depends on the states involved in the sequence.
When working with the groundrydberg
and digital
bases, atoms interact
when they are in the Rydberg state \(r\rangle\):
where \(\hat{n}_i = r\rangle\langle r_i\) (the projector of atom \(i\) onto the Rydberg state), \(R_{ij}^6\) is the distance between atoms \(i\) and \(j\) and \(C_6\) is a coefficient depending on the specific Rydberg level of \(r\rangle\).
On the other hand, with the two Rydberg states of the XY
basis, the interaction Hamiltonian takes the form
where \(C_3\) is a coefficient that depends on the chosen Ryberg states and
Note
The definitions given for both interaction Hamiltonians are independent of the chosen state vector convention.