Circuits¶
Conceptual overview¶
There are two primary representations of quantum programs in Cirq, each of
which are represented by a class: Circuit
and Schedule
.
Conceptually a Circuit object is very closely related to the
abstract quantum circuit model, while a Schedule object is
like the abstract quantum circuit model but includes detailed timing
information.
Conceptually: a Circuit
is a collection of Moments
. A
Moment
is a collection of Operations
that all act during
the same abstract time slice. An Operation
is a
some effect that operates on a specific subset of Qubits
,
the most common type of Operation
is a GateOperation
.
Circuits and Moments Let’s unpack this.
At the base of this construction is the notion of a qubit. In
Cirq, qubits and other quantum objects are identified by instances of subclasses
of the Qid
base class. Different subclasses of Qid
can be used
for different purposes. For example the qubits that Google’s
Xmon devices use are often arranged on the vertices of a
square grid. For this the class GridQubit
subclasses Qid
. For example, we can create
a 3 by 3 grid of qubits using
qubits = [cirq.GridQubit(x, y) for x in range(3) for y in range(3)]
print(qubits[0])
# prints "(0, 0)"
The next level up conceptually is the notion of a Gate
.
A Gate
represents a physical process that occurs on a
Qubit
. The important property of a Gate
is that it
can be applied on to one or more qubits. This can be done
via the Gate.on
method itself or via ()
and doing this
turns the Gate
into a GateOperation
.
# This is an Pauli X gate. It is an object instance.
x_gate = cirq.X
# Applying it to the qubit at location (0, 0) (defined above)
# turns it into an operation.
x_op = x_gate(qubits[0])
print(x_op)
# prints "X((0, 0))"
A Moment
is quite simply a collection of operations, each of
which operates on a different set of qubits, and which conceptually
represents these operations as occurring during this abstract time
slice. The Moment
structure itself is not required to be
related to the actual scheduling of the operations on a quantum
computer, or via a simulator, though it can be. For example, here
is a Moment in which Pauli X and a CZ gate operate on three qubits:
cz = cirq.CZ(qubits[0], qubits[1])
x = cirq.X(qubits[2])
moment = cirq.Moment([x, cz])
print(moment)
# prints "X((0, 2)) and CZ((0, 0), (0, 1))"
Note that is not the only way to construct moments, nor even the
typical method, but illustrates that a Moment
is just a
collection of operations on disjoint sets of qubits.
Finally at the top level a Circuit
is an ordered series
of Moment
s. The first Moment
in this series is,
conceptually, contains the first Operations
that will be
applied. Here, for example, is a simple circuit made up of
two moments
cz01 = cirq.CZ(qubits[0], qubits[1])
x2 = cirq.X(qubits[2])
cz12 = cirq.CZ(qubits[1], qubits[2])
moment0 = cirq.Moment([cz01, x2])
moment1 = cirq.Moment([cz12])
circuit = cirq.Circuit((moment0, moment1))
print(circuit)
# prints the text diagram for the circuit:
# (0, 0): ───@───────
# │
# (0, 1): ───@───@───
# │
# (0, 2): ───X───@───
Again, note that this is only one way to construct a Circuit
but illustrates the concept that a Circuit
is an iterable
of Moments
.
Constructing circuits¶
Constructing Circuits
as a series of Moments
with each
Moment
being handcrafted is tedious. Instead we provide a
variety of different manners to create a Circuit
.
One of the most useful ways to construct a Circuit
is by
appending onto the Circuit
with the
Circuit.append
method.
from cirq.ops import CZ, H
q0, q1, q2 = [cirq.GridQubit(i, 0) for i in range(3)]
circuit = cirq.Circuit()
circuit.append([CZ(q0, q1), H(q2)])
print(circuit)
# prints
# (0, 0): ───@───
# │
# (1, 0): ───@───
#
# (2, 0): ───H───
This appended an entire new moment to the qubit, which we can continue to do,
circuit.append([H(q0), CZ(q1, q2)])
print(circuit)
# prints
# (0, 0): ───@───H───
# │
# (1, 0): ───@───@───
# │
# (2, 0): ───H───@───
In these two examples, we have appending full moments, what happens when we append all of these at once?
circuit = cirq.Circuit()
circuit.append([CZ(q0, q1), H(q2), H(q0), CZ(q1, q2)])
print(circuit)
# prints
# (0, 0): ───@───H───
# │
# (1, 0): ───@───@───
# │
# (2, 0): ───H───@───
We see that here we have again created two Moments
. How did
Circuit
know how to do this? Circuit's
Circuit.append
method (and its cousin Circuit.insert
) both take an argument called
the InsertStrategy
. By default the InsertStrategy
is
InsertStrategy.NEW_THEN_INLINE
.
InsertStrategies¶
InsertStrategy
defines how Operations
are placed in a
Circuit
when requested to be inserted at a given location.
Here a location
is identified by the index of the Moment
(in
the Circuit
) where the insertion is requested to be placed at
(in the case of Circuit.append
this means inserting at the Moment
at an index one greater than the maximum moment index in the
Circuit
). There are four such strategies:
InsertStrategy.EARLIEST
, InsertStrategy.NEW
,
InsertStrategy.INLINE
and
InsertStrategy.NEW_THEN_INLINE
.
InsertStrategy.EARLIEST
is defined as
InsertStrategy.EARLIEST
: Scans backward from the insert location until a moment with operations touching qubits affected by the operation to insert is found. The operation is added into the moment just after that location.
For example, if we first create an Operation
in a single moment,
and then use InsertStrategy.EARLIEST
the Operation
can slide back to this
first Moment
if there is space:
from cirq.circuits import InsertStrategy
circuit = cirq.Circuit()
circuit.append([CZ(q0, q1)])
circuit.append([H(q0), H(q2)], strategy=InsertStrategy.EARLIEST)
print(circuit)
# prints
# (0, 0): ───@───H───
# │
# (1, 0): ───@───────
#
# (2, 0): ───H───────
After creating the first moment with a CZ
gate, the second
append uses the InsertStrategy.EARLIEST
strategy. The
H
on q0
cannot slide back, while the H
on
q2
can and so ends up in the first Moment
.
Contrast this with the InsertStrategy.NEW
InsertStrategy
:
InsertStrategy.NEW
: Every operation that is inserted is created in a new moment.
circuit = cirq.Circuit()
circuit.append([H(q0), H(q1), H(q2)], strategy=InsertStrategy.NEW)
print(circuit)
# prints
# (0, 0): ───H───────────
#
# (1, 0): ───────H───────
#
# (2, 0): ───────────H───
Here every operator processed by the append ends up in a new moment.
InsertStrategy.NEW
is most useful when you are inserting a single operation and
don’t want it to interfere with other Moments
.
Another strategy is InsertStrategy.INLINE
:
InsertStrategy.INLINE
: Attempts to add the operation to insert into the moment just before the desired insert location. But, if there’s already an existing operation affecting any of the qubits touched by the operation to insert, a new moment is created instead.
circuit = cirq.Circuit()
circuit.append([CZ(q1, q2)])
circuit.append([CZ(q1, q2)])
circuit.append([H(q0), H(q1), H(q2)], strategy=InsertStrategy.INLINE)
print(circuit)
# prints
# (0, 0): ───────H───────
#
# (1, 0): ───@───@───H───
# │ │
# (2, 0): ───@───@───H───
After two initial CZ
between the second and third qubit, we try
to insert 3 H
Operations
. We see that the H
on the first
qubit is inserted into the previous Moment
, but the H
on the second and third
qubits cannot be inserted into the previous Moment
, so a new Moment
is
created.
Finally we turn to the default strategy:
InsertStrategy.NEW_THEN_INLINE
: Creates a new moment at the desired insert location for the first operation, but then switches to inserting operations according toInsertStrategy.INLINE
.
circuit = cirq.Circuit()
circuit.append([H(q0)])
circuit.append([CZ(q1,q2), H(q0)], strategy=InsertStrategy.NEW_THEN_INLINE)
print(circuit)
# prints
# (0, 0): ───H───H───
#
# (1, 0): ───────@───
# │
# (2, 0): ───────@───
The first append creates a single moment with a H
on the first
qubit. Then the append with the InsertStrategy.NEW_THEN_INLINE
strategy begins by inserting the CZ
in a new
Moment
(the InsertStrategy.NEW
in
InsertStrategy.NEW_THEN_INLINE
). Subsequent appending is done
InsertStrategy.INLINE
so the next H
on the first
qubit is appending in the just created Moment
.
Here is a helpful diagram for the different InsertStrategies
.
TODO(dabacon): diagram.
Patterns for Arguments to Append and Insert¶
Above we have used a series of Circuit.append
calls with a list
of different Operations
we are adding to the circuit. But the argument
where we have supplied a list can also take more than just list
values.
Example:
def my_layer():
yield CZ(q0, q1)
yield [H(q) for q in (q0, q1, q2)]
yield [CZ(q1, q2)]
yield [H(q0), [CZ(q1, q2)]]
circuit = cirq.Circuit()
circuit.append(my_layer())
for x in my_layer():
print(x)
# prints
# CZ((0, 0), (1, 0))
# [cirq.H.on(cirq.GridQubit(0, 0)), cirq.H.on(cirq.GridQubit(1, 0)), cirq.H.on(cirq.GridQubit(2, 0))]
# [cirq.CZ.on(cirq.GridQubit(1, 0), cirq.GridQubit(2, 0))]
# [cirq.H.on(cirq.GridQubit(0, 0)), [cirq.CZ.on(cirq.GridQubit(1, 0), cirq.GridQubit(2, 0))]]
print(circuit)
# prints
# (0, 0): ───@───H───H───────
# │
# (1, 0): ───@───H───@───@───
# │ │
# (2, 0): ───H───────@───@───
Recall that in Python functions that have a yield
are generators.
Generators are functions that act as iterators. Above we see that we can
iterate over my_layer()
. We see that when we do this each of the
yields
produces what was yielded, and here these are Operations
,
lists of Operations
or lists of Operations
mixed with lists of
Operations
. But when we pass this iterator to the append method,
something magical happens. Circuit
is able to flatten all of
these an pass them as one giant list to Circuit.append
(this
also works for Circuit.insert
).
The above idea uses a concept we call anOP_TREE
. AnOP_TREE
is not a class, but a contract. The basic idea is that, if the input can be iteratively flattened into a list of operations, then the input is anOP_TREE
.
A very nice pattern emerges from this structure: define
generators for subcircuits, which can vary by size
or Operation
parameters.
Another useful method is to construct a Circuit
fully formed
from an OP_TREE
via the static method Circuit.from_ops
(which takes an insertion strategy as a parameter):
circuit = cirq.Circuit.from_ops(H(q0), H(q1))
print(circuit)
# prints
# (0, 0): ───H───
#
# (1, 0): ───H───
Slicing and Iterating over Circuits¶
Circuits
can be iterated over and sliced. When they are iterated
over each item in the iteration is a moment:
circuit = cirq.Circuit.from_ops(H(q0), CZ(q0, q1))
for moment in circuit:
print(moment)
# prints
# H((0, 0))
# CZ((0, 0), (1, 0))
Slicing a Circuit
on the other hand, produces a new
Circuit
with only the moments corresponding to the slice:
circuit = cirq.Circuit.from_ops(H(q0), CZ(q0, q1), H(q1), CZ(q0, q1))
print(circuit[1:3])
# prints
# (0, 0): ───@───────
# │
# (1, 0): ───@───H───
Especially useful is dropping the last moment (which are often just
measurements): circuit[:1]
, or reversing a circuit:
circuit[::1]
.
Importing from OpenQASM¶
The QASM importer is in experimental state and currently only supports a subset of the full OpenQASM spec.
Amongst others, classical control, arbitrary gate definitions and even some of the gates that don’t have a onetoone representation in Cirq are not yet supported.
The functionality should be sufficient to import interesting quantum circuits.
Error handling is very simple  on any lexical or syntactical error a QasmException
is raised.
Importing cirq.Circuit from QASM format¶
Requirements: ply
pip install ply==3.4
The following call will create a circuit defined by the input QASM string:
from cirq.contrib.qasm_import import circuit_from_qasm
circuit = circuit_from_qasm("""
OPENQASM 2.0;
include "qelib1.inc";
qreg q[3];
creg meas[3];
h q;
measure q > meas;
""")
print(circuit)
Results:
q_0: ───H───M('meas_0')───
q_1: ───H───M('meas_1')───
q_2: ───H───M('meas_2')───
Supported Control statements¶
Statement  Cirq support  Description  Example 

OPENQASM 2.0; 
supported  Denotes a file in Open QASM format  OPENQASM 2.0; 
qreg name[size]; 
supported (see mapping qubits)  Declare a named register of qubits  qreg q[5]; 
creg name[size]; 
supported (see mapping classical register to measurement keys)  Declare a named register of bits  creg c[5]; 
include "filename"; 
supported ONLY to include the standard "qelib1.inc" lib for compatibility  Open and parse another source file  include "qelib1.inc"; 
gate name(params) qargs; 
NOT supported  Declare a unitary gate  
opaque name(params) qargs; 
NOT supported  Declare an opaque gate  
// comment text 
supported  Comment a line of text  // supported! 
U(θ,φ,λ) qubit/qreg; 
supported  Apply builtin single qubit gate(s)  U(pi/2,2*pi/3,0) q[0]; 
CX qubit/qreg,qubit/qreg; 
supported  Apply builtin CNOT gate(s)  CX q[0],q[1]; 
measure qubit/qreg > bit/creg; 
supported  Make measurement(s) inZ basis 
measure q > c; 
reset qubit/qreg; 
NOT supported  Prepare qubit(s) in 0> 
reset q[0]; 
gatename(params) qargs; 
supported for ONLY the supported subset of standard gates defined in "qelib1.inc"  Apply a userdefined unitary gate  rz(pi/2) q[0]; 
if(creg==int) qop; 
NOT supported  Conditionally apply quantum operation  if(c==5) CX q[0],q[1]; 
barrier qargs; 
NOT supported  Prevent transformations across this source line  barrier q[0],q[1]; 
Gate conversion rules¶
Note: The standard Quantum Experience standard gates are defined in (qelib.inc)
based on the U
and CX
builtin instructions and we could generate them dynamically. Instead, we chose to map them to native cirq gates explicitly which results in a more user friendly circuit.
QE gates  Cirq translation  Notes 

U(θ,φ,λ) 
QasmUGate(θ,φ,λ) 

CX 
cirq.CX 

u3(θ,φ,λ) 
QasmUGate(θ,φ,λ) 

u2(φ,λ) = u3(π/2,φ,λ) 
QasmUGate(π/2,φ,λ) 

u1 (λ) = u3(0,0,λ) 
NOT supported  
id 
cirq.Identity 
one singlequbit Identity gate is created for each qubit if applied on a register 
u0(γ) 
NOT supported  this is the "WAIT gate" for length γ in QE 
x 
cirq.X 

y 
cirq.Y 

z 
cirq.Z 

h 
cirq.H 

s 
cirq.S 

sdg 
cirq.S**1 

t 
cirq.T 

tdg 
cirq.T**1 

rx(θ) 
cirq.Rx(θ) 

ry(θ) 
cirq.Ry(θ) 

rz(θ) 
cirq.Rz(θ) 

cx 
cirq.CX 

cy 
cirq.ControlledGate(cirq.Y) 

cz 
cirq.CZ 

ch 
cirq.ControlledGate(cirq.H) 

swap 
cirq.SWAP 

ccx 
cirq.CCX 

cswap 
cirq.CSWAP 

crz 
NOT supported  
cu1 
NOT supported  
cu3 
NOT supported  
rzz 
NOT supported 
Mapping quantum registers to qubits¶
For a quantum register qreg qfoo[n];
the QASM importer will create cirq.NamedQubit
s named qfoo_0
..qfoo_<n1>
.
Mapping classical registers to measurement keys¶
For a classical register creg cbar[n];
the QASM importer will create measurement keys named cbar_0
..cbar_<n1>
.