Expression of type Implies ¶

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, VertExprArray, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd
from proveit.logic import Implies
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import ket0, ket1, ket_plus
from proveit.physics.quantum.QPE import QPE1, _U, _ket_u, _phase, _psi_t_ket, _s
from proveit.physics.quantum.circuits import Gate, Input, MultiQubitElem, Output, Qcircuit, QcircuitEquiv

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(t, one)
sub_expr3 = Add(t, _s)
sub_expr4 = Interval(sub_expr2, sub_expr3)
sub_expr5 = Interval(one, sub_expr3)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE1(_U, t), part = sub_expr1), targets = sub_expr5)
sub_expr7 = MultiQubitElem(element = Output(state = TensorProd(_psi_t_ket, _ket_u), part = sub_expr1), targets = sub_expr5)
sub_expr8 = [ExprRange(sub_expr1, Input(state = ket_plus), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]
sub_expr9 = [ExprRange(sub_expr1, sub_expr6, one, t), ExprRange(sub_expr1, sub_expr6, sub_expr2, sub_expr3)]
sub_expr10 = [ExprRange(sub_expr1, sub_expr7, one, t), ExprRange(sub_expr1, sub_expr7, sub_expr2, sub_expr3).with_wrapping_at(2,6)]
sub_expr11 = [ExprRange(sub_expr1, Output(state = ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))), Add(Neg(t), one), zero).with_decreasing_order(), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]
expr = Implies(QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr11)), Qcircuit(vert_expr_array = VertExprArray(sub_expr10))), QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr8, sub_expr9, sub_expr11)), Qcircuit(vert_expr_array = VertExprArray(sub_expr8, sub_expr9, sub_expr10))).with_wrapping_at(1)).with_wrapping_at(2)

# 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(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
& \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
& \qout{\vdots} \\
& \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
& \qout{\lvert u \rangle}
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \multiqout{1}{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
& \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle}
} \end{array}\right)\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\vdots} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\lvert u \rangle}
} \end{array}\right) \\  \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}_1\left(U, t\right)} & \multiqout{4}{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle}
} \end{array}\right) \end{array} \end{array}\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.	()	(2)	('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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	10, 11
8	Operation	operator: 15 operand: 17
9	Operation	operator: 15 operand: 20
10	Operation	operator: 15 operands: 14
11	Operation	operator: 15 operands: 16
12	ExprTuple	17
13	ExprTuple	20
14	ExprTuple	18, 19, 17
15	Literal
16	ExprTuple	18, 19, 20
17	ExprTuple	21, 22
18	ExprTuple	23, 24
19	ExprTuple	25, 26
20	ExprTuple	27, 28
21	ExprRange	lambda_map: 29 start_index: 30 end_index: 101
22	ExprRange	lambda_map: 31 start_index: 112 end_index: 93
23	ExprRange	lambda_map: 32 start_index: 112 end_index: 104
24	ExprRange	lambda_map: 33 start_index: 112 end_index: 93
25	ExprRange	lambda_map: 34 start_index: 112 end_index: 104
26	ExprRange	lambda_map: 34 start_index: 73 end_index: 76
27	ExprRange	lambda_map: 35 start_index: 112 end_index: 104
28	ExprRange	lambda_map: 35 start_index: 73 end_index: 76
29	Lambda	parameter: 125 body: 36
30	Operation	operator: 85 operands: 37
31	Lambda	parameter: 125 body: 38
32	Lambda	parameter: 125 body: 39
33	Lambda	parameter: 125 body: 40
34	Lambda	parameter: 125 body: 41
35	Lambda	parameter: 125 body: 42
36	Operation	operator: 66 operands: 43
37	ExprTuple	44, 112
38	Operation	operator: 49 operands: 45
39	Operation	operator: 61 operands: 46
40	Operation	operator: 49 operands: 47
41	Operation	operator: 49 operands: 48
42	Operation	operator: 49 operands: 50
43	NamedExprs	state: 51
44	Operation	operator: 123 operand: 104
45	NamedExprs	element: 52 targets: 55
46	NamedExprs	state: 53
47	NamedExprs	element: 54 targets: 55
48	NamedExprs	element: 56 targets: 58
49	Literal
50	NamedExprs	element: 57 targets: 58
51	Operation	operator: 96 operands: 59
52	Operation	operator: 66 operands: 62
53	Operation	operator: 108 operand: 72
54	Operation	operator: 61 operands: 62
55	Operation	operator: 68 operands: 63
56	Operation	operator: 64 operands: 65
57	Operation	operator: 66 operands: 67
58	Operation	operator: 68 operands: 69
59	ExprTuple	70, 71
60	ExprTuple	72
61	Literal
62	NamedExprs	state: 92 part: 125
63	ExprTuple	73, 76
64	Literal
65	NamedExprs	operation: 74 part: 125
66	Literal
67	NamedExprs	state: 75 part: 125
68	Literal
69	ExprTuple	112, 76
70	Operation	operator: 105 operands: 77
71	Operation	operator: 78 operands: 79
72	Literal
73	Operation	operator: 85 operands: 80
74	Operation	operator: 81 operands: 82
75	Operation	operator: 83 operands: 84
76	Operation	operator: 85 operands: 86
77	ExprTuple	112, 87
78	Literal
79	ExprTuple	88, 89
80	ExprTuple	104, 112
81	Literal
82	ExprTuple	90, 104
83	Literal
84	ExprTuple	91, 92
85	Literal
86	ExprTuple	104, 93
87	Operation	operator: 119 operands: 94
88	Operation	operator: 108 operand: 101
89	Operation	operator: 96 operands: 97
90	Literal
91	Operation	operator: 98 operand: 104
92	Literal
93	Literal
94	ExprTuple	121, 100
95	ExprTuple	101
96	Literal
97	ExprTuple	102, 103
98	Literal
99	ExprTuple	104
100	Operation	operator: 105 operands: 106
101	Literal
102	Operation	operator: 119 operands: 107
103	Operation	operator: 108 operand: 112
104	Variable
105	Literal
106	ExprTuple	112, 121
107	ExprTuple	110, 111
108	Literal
109	ExprTuple	112
110	Literal
111	Operation	operator: 113 operands: 114
112	Literal
113	Literal
114	ExprTuple	121, 115, 116, 117, 118
115	Literal
116	Literal
117	Operation	operator: 119 operands: 120
118	Literal
119	Literal
120	ExprTuple	121, 122
121	Literal
122	Operation	operator: 123 operand: 125
123	Literal
124	ExprTuple	125
125	Variable

Expression of type Implies¶

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

Expression of type Implies ¶

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