Expression of type Equals ¶

from the theory of proveit.physics.quantum.QPE ¶

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import ExprRange, Variable
from proveit.core_expr_types import Len
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals
from proveit.numbers import Add, Interval, Mult, one, three
from proveit.physics.quantum import I
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _Psi_ket, _ket_u, _psi__t_ket, _s, _t
from proveit.physics.quantum.circuits import Gate, Input, MultiQubitElem, Output

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(_t, _s)
sub_expr3 = Interval(one, _t)
expr = Equals(Len(operands = [ExprRange(sub_expr1, MultiQubitElem(element = Input(state = TensorProd(_psi__t_ket, _ket_u), part = sub_expr1), targets = Interval(one, sub_expr2)), one, sub_expr2), ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(_t), part = sub_expr1), targets = sub_expr3), one, _t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, _s), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _Psi_ket, part = sub_expr1), targets = sub_expr3), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = Interval(Add(_t, one), sub_expr2)), one, _s)]), Add(Mult(three, _t), Mult(three, _s))).with_wrapping_at(1)

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\begin{array}{c} \begin{array}{l} |\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~t + s~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, ..\left(s - 3\right) \times.., \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}\right)| \\  = \left(\left(3 \cdot t\right) + \left(3 \cdot s\right)\right) \end{array} \end{array}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(1)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

	core type	sub-expressions
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operands: 6
4	Operation	operator: 65 operands: 7
5	Literal
6	ExprTuple	8, 9, 10, 11, 12
7	ExprTuple	13, 14
8	ExprRange	lambda_map: 15 start_index: 69 end_index: 60
9	ExprRange	lambda_map: 16 start_index: 69 end_index: 73
10	ExprRange	lambda_map: 17 start_index: 69 end_index: 70
11	ExprRange	lambda_map: 18 start_index: 69 end_index: 73
12	ExprRange	lambda_map: 19 start_index: 69 end_index: 70
13	Operation	operator: 21 operands: 20
14	Operation	operator: 21 operands: 22
15	Lambda	parameter: 58 body: 23
16	Lambda	parameter: 58 body: 24
17	Lambda	parameter: 58 body: 25
18	Lambda	parameter: 58 body: 26
19	Lambda	parameter: 58 body: 28
20	ExprTuple	29, 73
21	Literal
22	ExprTuple	29, 70
23	Operation	operator: 34 operands: 30
24	Operation	operator: 34 operands: 31
25	Operation	operator: 47 operands: 32
26	Operation	operator: 34 operands: 33
27	ExprTuple	58
28	Operation	operator: 34 operands: 35
29	Literal
30	NamedExprs	element: 36 targets: 37
31	NamedExprs	element: 38 targets: 41
32	NamedExprs	operation: 39
33	NamedExprs	element: 40 targets: 41
34	Literal
35	NamedExprs	element: 42 targets: 43
36	Operation	operator: 44 operands: 45
37	Operation	operator: 53 operands: 46
38	Operation	operator: 47 operands: 48
39	Literal
40	Operation	operator: 51 operands: 49
41	Operation	operator: 53 operands: 50
42	Operation	operator: 51 operands: 52
43	Operation	operator: 53 operands: 54
44	Literal
45	NamedExprs	state: 55 part: 58
46	ExprTuple	69, 60
47	Literal
48	NamedExprs	operation: 56 part: 58
49	NamedExprs	state: 57 part: 58
50	ExprTuple	69, 73
51	Literal
52	NamedExprs	state: 68 part: 58
53	Literal
54	ExprTuple	59, 60
55	Operation	operator: 61 operands: 62
56	Operation	operator: 63 operand: 73
57	Literal
58	Variable
59	Operation	operator: 65 operands: 64
60	Operation	operator: 65 operands: 66
61	Literal
62	ExprTuple	67, 68
63	Literal
64	ExprTuple	73, 69
65	Literal
66	ExprTuple	73, 70
67	Operation	operator: 71 operand: 73
68	Literal
69	Literal
70	Literal
71	Literal
72	ExprTuple	73
73	Literal

Expression of type Equals¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Equals ¶

from the theory of proveit.physics.quantum.QPE ¶