/
rbtheory.py
868 lines (724 loc) · 38 KB
/
rbtheory.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
"""
RB-related functions of gates and models
"""
#***************************************************************************************************
# Copyright 2015, 2019 National Technology & Engineering Solutions of Sandia, LLC (NTESS).
# Under the terms of Contract DE-NA0003525 with NTESS, the U.S. Government retains certain rights
# in this software.
# Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except
# in compliance with the License. You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0 or in the LICENSE file in the root pyGSTi directory.
#***************************************************************************************************
from . import rbtools as _rbtls
from . import optools as _optls
from . import matrixtools as _mtls
from .. import objects as _objs
#from .. import construction as _cnst
#from .. import algorithms as _algs
import numpy as _np
import warnings as _warnings
def predicted_rb_number(model, target_model, weights=None, d=None, rtype='EI'):
"""
Predicts the RB error rate from a model.
Uses the "L-matrix" theory from Proctor et al Phys. Rev. Lett. 119, 130502
(2017). Note that this gives the same predictions as the theory in Wallman
Quantum 2, 47 (2018).
This theory is valid for various types of RB, including standard
Clifford RB -- i.e., it will accurately predict the per-Clifford
error rate reported by standard Clifford RB. It is also valid for
"direct RB" under broad circumstances.
For this function to be valid the model should be trace preserving
and completely positive in some representation, but the particular
representation of the model used is irrelevant, as the predicted RB
error rate is a gauge-invariant quantity. The function is likely reliable
when complete positivity is slightly violated, although the theory on
which it is based assumes complete positivity.
Parameters
----------
model : Model
The model to calculate the RB number of. This model is the
model randomly sampled over, so this is not necessarily the
set of physical primitives. In Clifford RB this is a set of
Clifford gates; in "direct RB" this normally would be the
physical primitives.
target_model : Model
The target model, corresponding to `model`. This function is not invariant
under swapping `model` and `target_model`: this Model must be the target model,
and should consistent of perfect gates.
weights : dict, optional
If not None, a dictionary of floats, whereby the keys are the gates
in `model` and the values are the unnormalized probabilities to apply
each gate at each stage of the RB protocol. If not None, the values
in weights must all be non-negative, and they must not all be zero.
Because, when divided by their sum, they must be a valid probability
distribution. If None, the weighting defaults to an equal weighting
on all gates, as this is used in many RB protocols (e.g., Clifford RB).
But, this weighting is flexible in the "direct RB" protocol.
d : int, optional
The Hilbert space dimension. If None, then sqrt(model.dim) is used.
rtype : str, optional
The type of RB error rate, either "EI" or "AGI", corresponding to
different dimension-dependent rescalings of the RB decay constant
p obtained from fitting to Pm = A + Bp^m. "EI" corresponds to
an RB error rate that is associated with entanglement infidelity, which
is the probability of error for a gate with stochastic errors. This is
the RB error rate defined in the "direct RB" protocol, and is given by:
r = (d^2 - 1)(1 - p)/d^2,
The AGI-type r is given by
r = (d - 1)(1 - p)/d,
which is the conventional r definition in Clifford RB. This r is
associated with (gate-averaged) average gate infidelity.
Returns
-------
r : float.
The predicted RB number.
"""
if d is None: d = int(round(_np.sqrt(model.dim)))
p = predicted_rb_decay_parameter(model, target_model, weights=weights)
r = _rbtls.p_to_r(p, d=d, rtype=rtype)
return r
def predicted_rb_decay_parameter(model, target_model, weights=None):
"""
Computes the second largest eigenvalue of the 'L matrix' (see the `L_matrix` function).
For standard Clifford RB and direct RB, this corresponds to the RB decay
parameter p in Pm = A + Bp^m for "reasonably low error" trace preserving and
completely positive gates. See also the `predicted_rb_number` function.
Parameters
----------
model : Model
The model to calculate the RB decay parameter of. This model is the
model randomly sampled over, so this is not necessarily the
set of physical primitives. In Clifford RB this is a set of
Clifford gates; in "direct RB" this normally would be the
physical primitives.
target_model : Model
The target model corresponding to model. This function is not invariant under
swapping `model` and `target_model`: this Model must be the target model, and
should consistent of perfect gates.
weights : dict, optional
If not None, a dictionary of floats, whereby the keys are the gates
in `model` and the values are the unnormalized probabilities to apply
each gate at each stage of the RB protocol. If not None, the values
in weights must all be non-negative, and they must not all be zero.
Because, when divided by their sum, they must be a valid probability
distribution. If None, the weighting defaults to an equal weighting
on all gates, as this is used in many RB protocols (e.g., Clifford RB).
But, this weighting is flexible in the "direct RB" protocol.
Returns
-------
p : float.
The second largest eigenvalue of L. This is the RB decay parameter
for various types of RB.
"""
L = L_matrix(model, target_model, weights=weights)
E = _np.absolute(_np.linalg.eigvals(L))
E = _np.flipud(_np.sort(E))
if abs(E[0] - 1) > 10**(-12):
_warnings.warn("Output may be unreliable because the model is not approximately trace-preserving.")
if E[1].imag > 10**(-10):
_warnings.warn("Output may be unreliable because the RB decay constant has a significant imaginary component.")
p = abs(E[1])
return p
def rb_gauge(model, target_model, weights=None, mx_basis=None, eigenvector_weighting=1.0):
"""
Computes the gauge transformation required so that the RB number matches the average model infidelity.
This function computes the gauge transformation required so that, when the
model is transformed via this gauge-transformation, the RB number -- as
predicted by the function `predicted_rb_number` -- is the average model
infidelity between the transformed `model` model and the target model
`target_model`. This transformation is defined Proctor et al
Phys. Rev. Lett. 119, 130502 (2017), and see also Wallman Quantum 2, 47
(2018).
Parameters
----------
model : Model
The RB model. This is not necessarily the set of physical primitives -- it
is the model randomly sampled over in the RB protocol (e.g., the Cliffords).
target_model : Model
The target model corresponding to model. This function is not invariant under
swapping `model` and `target_model`: this Model must be the target model, and
should consistent of perfect gates.
weights : dict, optional
If not None, a dictionary of floats, whereby the keys are the gates
in `model` and the values are the unnormalized probabilities to apply
each gate at each stage of the RB protocol. If not None, the values
in weights must all be non-negative, and they must not all be zero.
Because, when divided by their sum, they must be a valid probability
distribution. If None, the weighting defaults to an equal weighting
on all gates, as this is used in many RB protocols (e.g., Clifford RB).
But, this weighting is flexible in the "direct RB" protocol.
mx_basis : {"std","gm","pp"}, optional
The basis of the models. If None, the basis is obtained from the model.
eigenvector_weighting : float, optional
Must be non-zero. A weighting on the eigenvector with eigenvalue that
is the RB decay parameter, in the sum of this eigenvector and the
eigenvector with eigenvalue of 1 that defines the returned matrix `l_operator`.
The value of this factor does not change whether this `l_operator` transforms into
a gauge in which r = AGsI, but it may impact on other properties of the
gates in that gauge. It is irrelevant if the gates are unital.
Returns
-------
l_operator : array
The matrix defining the gauge-transformation.
"""
gam, vecs = _np.linalg.eig(L_matrix(model, target_model, weights=weights))
absgam = abs(gam)
index_max = _np.argmax(absgam)
gam_max = gam[index_max]
if abs(gam_max - 1) > 10**(-12):
_warnings.warn("Output may be unreliable because the model is not approximately trace-preserving.")
absgam[index_max] = 0.0
index_2ndmax = _np.argmax(absgam)
decay_constant = gam[index_2ndmax]
if decay_constant.imag > 10**(-12):
_warnings.warn("Output may be unreliable because the RB decay constant has a significant imaginary component.")
vec_l_operator = vecs[:, index_max] + eigenvector_weighting * vecs[:, index_2ndmax]
if mx_basis is None:
mx_basis = model.basis.name
assert(mx_basis == 'pp' or mx_basis == 'gm' or mx_basis == 'std'), "mx_basis must be 'gm', 'pp' or 'std'."
if mx_basis in ('pp', 'gm'):
assert(_np.amax(vec_l_operator.imag) < 10**(-15)), "If 'gm' or 'pp' basis, RB gauge matrix should be real."
vec_l_operator = vec_l_operator.real
vec_l_operator[abs(vec_l_operator) < 10**(-15)] = 0.
l_operator = _mtls.unvec(vec_l_operator)
return l_operator
def transform_to_rb_gauge(model, target_model, weights=None, mx_basis=None, eigenvector_weighting=1.0):
"""
Transforms a Model into the "RB gauge" (see the `RB_gauge` function).
This notion was introduced in Proctor et al Phys. Rev. Lett. 119, 130502
(2017). This gauge is a function of both the model and its target. These may
be input in any gauge, for the purposes of obtaining "r = average model
infidelity" between the output :class:`Model` and `target_model`.
Parameters
----------
model : Model
The RB model. This is not necessarily the set of physical primitives -- it
is the model randomly sampled over in the RB protocol (e.g., the Cliffords).
target_model : Model
The target model corresponding to model. This function is not invariant under
swapping `model` and `target_model`: this Model must be the target model, and
should consistent of perfect gates.
weights : dict, optional
If not None, a dictionary of floats, whereby the keys are the gates
in `model` and the values are the unnormalized probabilities to apply
each gate at each stage of the RB protocol. If not None, the values
in weights must all be non-negative, and they must not all be zero.
Because, when divided by their sum, they must be a valid probability
distribution. If None, the weighting defaults to an equal weighting
on all gates, as this is used in many RB protocols (e.g., Clifford RB).
But, this weighting is flexible in the "direct RB" protocol.
mx_basis : {"std","gm","pp"}, optional
The basis of the models. If None, the basis is obtained from the model.
eigenvector_weighting : float, optional
Must be non-zero. A weighting on the eigenvector with eigenvalue that
is the RB decay parameter, in the sum of this eigenvector and the
eigenvector with eigenvalue of 1 that defines the returned matrix `l_operator`.
The value of this factor does not change whether this `l_operator` transforms into
a gauge in which r = AGsI, but it may impact on other properties of the
gates in that gauge. It is irrelevant if the gates are unital.
Returns
-------
model_in_RB_gauge : Model
The model `model` transformed into the "RB gauge".
"""
l = rb_gauge(model, target_model, weights=weights, mx_basis=mx_basis,
eigenvector_weighting=eigenvector_weighting)
model_in_RB_gauge = model.copy()
S = _objs.FullGaugeGroupElement(_np.linalg.inv(l))
model_in_RB_gauge.transform_inplace(S)
return model_in_RB_gauge
def L_matrix(model, target_model, weights=None): # noqa N802
"""
Constructs a generalization of the 'L-matrix' linear operator on superoperators.
From Proctor et al Phys. Rev. Lett. 119, 130502 (2017), the 'L-matrix' is
represented as a matrix via the "stack" operation. This eigenvalues of this
matrix describe the decay constant (or constants) in an RB decay curve for
an RB protocol whereby random elements of the provided model are sampled
according to the `weights` probability distribution over the model. So, this
facilitates predictions of Clifford RB and direct RB decay curves.
Parameters
----------
model : Model
The RB model. This is not necessarily the set of physical primitives -- it
is the model randomly sampled over in the RB protocol (e.g., the Cliffords).
target_model : Model
The target model corresponding to model. This function is not invariant under
swapping `model` and `target_model`: this Model must be the target model, and
should consistent of perfect gates.
weights : dict, optional
If not None, a dictionary of floats, whereby the keys are the gates
in `model` and the values are the unnormalized probabilities to apply
each gate at each stage of the RB protocol. If not None, the values
in weights must all be non-negative, and they must not all be zero.
Because, when divided by their sum, they must be a valid probability
distribution. If None, the weighting defaults to an equal weighting
on all gates, as this is used in many RB protocols (e.g., Clifford RB).
But, this weighting is flexible in the "direct RB" protocol.
Returns
-------
L : float
A weighted version of the L operator from Proctor et al Phys. Rev. Lett.
119, 130502 (2017), represented as a matrix using the 'stacking' convention.
"""
if weights is None:
weights = {}
for key in list(target_model.operations.keys()):
weights[key] = 1.
normalizer = _np.sum(_np.array([weights[key] for key in list(target_model.operations.keys())]))
L_matrix = (1 / normalizer) * _np.sum(
weights[key] * _np.kron(
model.operations[key].to_dense().T, _np.linalg.inv(target_model.operations[key].to_dense())
) for key in target_model.operations.keys())
return L_matrix
def R_matrix_predicted_rb_decay_parameter(model, group, group_to_model=None, weights=None): # noqa N802
"""
Returns the second largest eigenvalue of a generalization of the 'R-matrix' [see the `R_matrix` function].
Introduced in Proctor et al Phys. Rev. Lett. 119, 130502 (2017). This
number is a prediction of the RB decay parameter for trace-preserving gates
and a variety of forms of RB, including Clifford and direct RB. This
function creates a matrix which scales super-exponentially in the number of
qubits.
Parameters
----------
model : Model
The model to predict the RB decay paramter for. If `group_to_model` is
None, the labels of the gates in `model` should be the same as the labels of the
group elements in `group`. For Clifford RB this would be the clifford model,
for direct RB it would be the primitive gates.
group : MatrixGroup
The group that the `model` model contains gates from (`model` does not
need to be the full group, and could be a subset of `group`). For
Clifford RB and direct RB, this would be the Clifford group.
group_to_model : dict, optional
If not None, a dictionary that maps labels of group elements to labels
of `model`. If `model` and `group` elements have the same labels, this dictionary
is not required. Otherwise it is necessary.
weights : dict, optional
If not None, a dictionary of floats, whereby the keys are the gates in `model`
and the values are the unnormalized probabilities to apply each gate at
each stage of the RB protocol. If not None, the values in weights must all
be positive or zero, and they must not all be zero (because, when divided by
their sum, they must be a valid probability distribution). If None, the
weighting defaults to an equal weighting on all gates, as used in most RB
protocols.
Returns
-------
p : float
The predicted RB decay parameter. Valid for standard Clifford RB or direct RB
with trace-preserving gates, and in a range of other circumstances.
"""
R = R_matrix(model, group, group_to_model=group_to_model, weights=weights)
E = _np.absolute(_np.linalg.eigvals(R))
E = _np.flipud(_np.sort(E))
p = E[1]
return p
def R_matrix(model, group, group_to_model=None, weights=None): # noqa N802
"""
Constructs a generalization of the 'R-matrix' of Proctor et al Phys. Rev. Lett. 119, 130502 (2017).
This matrix described the exact behaviour of the average success
probablities of RB sequences. This matrix is super-exponentially large in
the number of qubits, but can be constructed for 1-qubit models.
Parameters
----------
model : Model
The noisy model (e.g., the Cliffords) to calculate the R matrix of.
The correpsonding `target` model (not required in this function)
must be equal to or a subset of (a faithful rep of) the group `group`.
If `group_to_model `is None, the labels of the gates in model should be
the same as the labels of the corresponding group elements in `group`.
For Clifford RB `model` should be the clifford model; for direct RB
this should be the native model.
group : MatrixGroup
The group that the `model` model contains gates from. For Clifford RB
or direct RB, this would be the Clifford group.
group_to_model : dict, optional
If not None, a dictionary that maps labels of group elements to labels
of model. This is required if the labels of the gates in `model` are different
from the labels of the corresponding group elements in `group`.
weights : dict, optional
If not None, a dictionary of floats, whereby the keys are the gates in model
and the values are the unnormalized probabilities to apply each gate at
for each layer of the RB protocol. If None, the weighting defaults to an
equal weighting on all gates, as used in most RB protocols (e.g., Clifford
RB).
Returns
-------
R : float
A weighted, a subset-sampling generalization of the 'R-matrix' from Proctor
et al Phys. Rev. Lett. 119, 130502 (2017).
"""
if group_to_model is None:
for key in list(model.operations.keys()):
assert(key in group.labels), "Gates labels are not in `group`!"
else:
for key in list(model.operations.keys()):
assert(key in group_to_model.values()), "Gates labels are not in `group_to_model`!"
d = int(round(_np.sqrt(model.dim)))
group_dim = len(group)
R_dim = group_dim * d**2
R = _np.zeros([R_dim, R_dim], float)
if weights is None:
weights = {}
for key in list(model.operations.keys()):
weights[key] = 1.
normalizer = _np.sum(_np.array([weights[key] for key in list(model.operations.keys())]))
for i in range(0, group_dim):
for j in range(0, group_dim):
label_itoj = group.labels[group.product([group.inverse_index(i), j])]
if group_to_model is not None:
if label_itoj in group_to_model:
gslabel = group_to_model[label_itoj]
R[j * d**2:(j + 1) * d**2, i * d**2:(i + 1) * d**2] = weights[gslabel] * model.operations[gslabel]
else:
if label_itoj in list(model.operations.keys()):
gslabel = label_itoj
R[j * d**2:(j + 1) * d**2, i * d**2:(i + 1) * d**2] = weights[gslabel] * model.operations[gslabel]
R = R / normalizer
return R
### COMMENTED OUT SO THAT THIS FILE DOESN'T NEED "from .. import construction as _cnst".
### THIS SHOULD BE ADDED BACK IN AT SOME POINT.
# def exact_rb_asps(model, group, m_max, m_min=0, m_step=1, success_outcomelabel=('0',),
# group_to_model=None, weights=None, compilation=None, group_twirled=False):
# """
# Calculates the exact RB average success probablilites (ASP).
# Uses some generalizations of the formula given Proctor et al
# Phys. Rev. Lett. 119, 130502 (2017). This formula does not scale well with
# group size and qubit number, and for the Clifford group it is likely only
# practical for a single qubit.
# Parameters
# ----------
# model : Model
# The noisy model (e.g., the Cliffords) to calculate the R matrix of.
# The correpsonding `target` model (not required in this function)
# must be equal to or a subset of (a faithful rep of) the group `group`.
# If group_to_model is None, the labels of the gates in model should be
# the same as the labels of the corresponding group elements in `group`.
# For Clifford RB `model` should be the clifford model; for direct RB
# this should be the native model.
# group : MatrixGroup
# The group that the `model` model contains gates from. For Clifford RB
# or direct RB, this would be the Clifford group.
# m_max : int
# The maximal sequence length of the random gates, not including the
# inversion gate.
# m_min : int, optional
# The minimal sequence length. Defaults to the smallest valid value of 0.
# m_step : int, optional
# The step size between sequence lengths. Defaults to the smallest valid
# value of 1.
# success_outcomelabel : str or tuple, optional
# The outcome label associated with success.
# group_to_model : dict, optional
# If not None, a dictionary that maps labels of group elements to labels
# of model. This is required if the labels of the gates in `model` are different
# from the labels of the corresponding group elements in `group`.
# weights : dict, optional
# If not None, a dictionary of floats, whereby the keys are the gates in model
# and the values are the unnormalized probabilities to apply each gate at
# for each layer of the RB protocol. If None, the weighting defaults to an
# equal weighting on all gates, as used in most RB protocols (e.g., Clifford
# RB).
# compilation : dict, optional
# If `model` is not the full group `group` (with the same labels), then a
# compilation for the group elements, used to implement the inversion gate
# (and the initial randomgroup element, if `group_twirled` is True). This
# is a dictionary with the group labels as keys and a gate sequence of the
# elements of `model` as values.
# group_twirled : bool, optional
# If True, the random sequence starts with a single uniformly random group
# element before the m random elements of `model`.
# Returns
# -------
# m : float
# Array of sequence length values that the ASPs have been calculated for.
# P_m : float
# Array containing ASP values for the specified sequence length values.
# """
# if compilation is None:
# for key in list(model.operations.keys()):
# assert(key in group.labels), "Gates labels are not in `group`, so `compilation must be specified."
# for label in group.labels:
# assert(label in list(model.operations.keys())
# ), "Some group elements not in `model`, so `compilation must be specified."
# i_max = _np.floor((m_max - m_min) / m_step).astype('int')
# m = _np.zeros(1 + i_max, int)
# P_m = _np.zeros(1 + i_max, float)
# group_dim = len(group)
# R = R_matrix(model, group, group_to_model=group_to_model, weights=weights)
# success_prepLabel = list(model.preps.keys())[0] # just take first prep
# success_effectLabel = success_outcomelabel[-1] if isinstance(success_outcomelabel, tuple) \
# else success_outcomelabel
# extended_E = _np.kron(_mtls.column_basis_vector(0, group_dim).T, model.povms['Mdefault'][success_effectLabel].T)
# extended_rho = _np.kron(_mtls.column_basis_vector(0, group_dim), model.preps[success_prepLabel])
# if compilation is None:
# extended_E = group_dim * _np.dot(extended_E, R)
# if group_twirled is True:
# extended_rho = _np.dot(R, extended_rho)
# else:
# full_model = _cnst.create_explicit_alias_model(model, compilation)
# R_fullgroup = R_matrix(full_model, group)
# extended_E = group_dim * _np.dot(extended_E, R_fullgroup)
# if group_twirled is True:
# extended_rho = _np.dot(R_fullgroup, extended_rho)
# Rstep = _np.linalg.matrix_power(R, m_step)
# Riterate = _np.linalg.matrix_power(R, m_min)
# for i in range(0, 1 + i_max):
# m[i] = m_min + i * m_step
# P_m[i] = _np.dot(extended_E, _np.dot(Riterate, extended_rho))
# Riterate = _np.dot(Rstep, Riterate)
# return m, P_m
### COMMENTED OUT SO THAT THIS FILE DOESN'T NEED "from .. import construction as _cnst"
### THIS SHOULD BE ADDED BACK IN AT SOME POINT.
# def L_matrix_asps(model, target_model, m_max, m_min=0, m_step=1, success_outcomelabel=('0',), # noqa N802
# compilation=None, group_twirled=False, weights=None, gauge_optimize=True,
# return_error_bounds=False, norm='diamond'):
# """
# Computes RB average survival probablities, as predicted by the 'L-matrix' theory.
# This theory was introduced in Proctor et al Phys. Rev. Lett. 119, 130502
# (2017). Within the function, the model is gauge-optimized to target_model. This is
# *not* optimized to the gauge specified by Proctor et al, but instead performs the
# standard pyGSTi gauge-optimization (using the frobenius distance). In most cases,
# this is likely to be a reasonable proxy for the gauge optimization perscribed by
# Proctor et al.
# Parameters
# ----------
# model : Model
# The noisy model.
# target_model : Model
# The target model.
# m_max : int
# The maximal sequence length of the random gates, not including the inversion gate.
# m_min : int, optional
# The minimal sequence length. Defaults to the smallest valid value of 0.
# m_step : int, optional
# The step size between sequence lengths.
# success_outcomelabel : str or tuple, optional
# The outcome label associated with success.
# compilation : dict, optional
# If `model` is not the full group, then a compilation for the group elements,
# used to implement the inversion gate (and the initial random group element,
# if `group_twirled` is True). This is a dictionary with the group labels as
# keys and a gate sequence of the elements of `model` as values.
# group_twirled : bool, optional
# If True, the random sequence starts with a single uniformly random group
# element before the m random elements of `model`.
# weights : dict, optional
# If not None, a dictionary of floats, whereby the keys are the gates in model
# and the values are the unnormalized probabilities to apply each gate at
# for each layer of the RB protocol. If None, the weighting defaults to an
# equal weighting on all gates, as used in most RB protocols (e.g., Clifford
# RB).
# gauge_optimize : bool, optional
# If True a gauge-optimization to the target model is implemented before
# calculating all quantities. If False, no gauge optimization is performed.
# Whether or not a gauge optimization is performed does not affect the rate of
# decay but it will generally affect the exact form of the decay. E.g., if a
# perfect model is given to the function -- but in the "wrong" gauge -- no
# decay will be observed in the output P_m, but the P_m can be far from 1 (even
# for perfect SPAM) for all m. The gauge optimization is optional, as it is
# not guaranteed to always improve the accuracy of the reported P_m, although when
# gauge optimization is performed this limits the possible deviations of the
# reported P_m from the true P_m.
# return_error_bounds : bool, optional
# Sets whether or not to return error bounds for how far the true ASPs can deviate
# from the values returned by this function.
# norm : str, optional
# The norm used in the error bound calculation. Either 'diamond' for the diamond
# norm (the default) or '1to1' for the Hermitian 1 to 1 norm.
# Returns
# -------
# m : float
# Array of sequence length values that the ASPs have been calculated for.
# P_m : float
# Array containing predicted ASP values for the specified sequence length values.
# if error_bounds is True :
# lower_bound: float
# Array containing lower bounds on the possible ASP values
# upper_bound: float
# Array containing upper bounds on the possible ASP values
# """
# d = int(round(_np.sqrt(model.dim)))
# if gauge_optimize:
# model_go = _algs.gaugeopt_to_target(model, target_model)
# else:
# model_go = model.copy()
# L = L_matrix(model_go, target_model, weights=weights)
# success_prepLabel = list(model.preps.keys())[0] # just take first prep
# success_effectLabel = success_outcomelabel[-1] if isinstance(success_outcomelabel, tuple) \
# else success_outcomelabel
# identity_vec = _mtls.vec(_np.identity(d**2, float))
# if compilation is not None:
# model_group = _cnst.create_explicit_alias_model(model_go, compilation)
# model_target_group = _cnst.create_explicit_alias_model(target_model, compilation)
# delta = gate_dependence_of_errormaps(model_group, model_target_group, norm=norm)
# emaps = errormaps(model_group, model_target_group)
# E_eff = _np.dot(model_go.povms['Mdefault'][success_effectLabel].T, emaps.operations['Gavg'])
# if group_twirled is True:
# L_group = L_matrix(model_group, model_target_group)
# if compilation is None:
# delta = gate_dependence_of_errormaps(model_go, target_model, norm=norm)
# emaps = errormaps(model_go, target_model)
# E_eff = _np.dot(model_go.povms['Mdefault'][success_effectLabel].T, emaps.operations['Gavg'])
# i_max = _np.floor((m_max - m_min) / m_step).astype('int')
# m = _np.zeros(1 + i_max, int)
# P_m = _np.zeros(1 + i_max, float)
# upper_bound = _np.zeros(1 + i_max, float)
# lower_bound = _np.zeros(1 + i_max, float)
# Lstep = _np.linalg.matrix_power(L, m_step)
# Literate = _np.linalg.matrix_power(L, m_min)
# for i in range(0, 1 + i_max):
# m[i] = m_min + i * m_step
# if group_twirled:
# L_m_rdd = _mtls.unvec(_np.dot(L_group, _np.dot(Literate, identity_vec)))
# else:
# L_m_rdd = _mtls.unvec(_np.dot(Literate, identity_vec))
# P_m[i] = _np.dot(E_eff, _np.dot(L_m_rdd, model_go.preps[success_prepLabel]))
# Literate = _np.dot(Lstep, Literate)
# upper_bound[i] = P_m[i] + delta / 2
# lower_bound[i] = P_m[i] - delta / 2
# if upper_bound[i] > 1:
# upper_bound[i] = 1.
# if lower_bound[i] < 0:
# lower_bound[i] = 0.
# if return_error_bounds:
# return m, P_m, lower_bound, upper_bound
# else:
# return m, P_m
def errormaps(model, target_model):
"""
Computes the 'left-multiplied' error maps associated with a noisy gate set, along with the average error map.
This is the model [E_1,...] such that
`G_i = E_iT_i`,
where `T_i` is the gate which `G_i` is a noisy
implementation of. There is an additional gate in the set, that has
the key 'Gavg'. This is the average of the error maps.
Parameters
----------
model : Model
The imperfect model.
target_model : Model
The target model.
Returns
-------
errormaps : Model
The left multplied error gates, along with the average error map,
with the key 'Gavg'.
"""
errormaps_gate_list = []
errormaps = model.copy()
for gate in list(target_model.operations.keys()):
errormaps.operations[gate] = _np.dot(model.operations[gate],
_np.transpose(target_model.operations[gate]))
errormaps_gate_list.append(errormaps.operations[gate])
errormaps.operations['Gavg'] = _np.mean(_np.array([i for i in errormaps_gate_list]),
axis=0, dtype=_np.float64)
return errormaps
def gate_dependence_of_errormaps(model, target_model, norm='diamond', mx_basis=None):
"""
Computes the "gate-dependence of errors maps" parameter defined by
delta_avg = avg_i|| E_i - avg_i(E_i) ||,
where E_i are the error maps, and the norm is either the diamond norm
or the 1-to-1 norm. This quantity is defined in Magesan et al PRA 85
042311 2012.
Parameters
----------
model : Model
The actual model
target_model : Model
The target model.
norm : str, optional
The norm used in the calculation. Can be either 'diamond' for
the diamond norm, or '1to1' for the Hermitian 1 to 1 norm.
mx_basis : {"std","gm","pp"}, optional
The basis of the models. If None, the basis is obtained from
the model.
Returns
-------
delta_avg : float
The value of the parameter defined above.
"""
error_gs = errormaps(model, target_model)
delta = []
if mx_basis is None:
mx_basis = model.basis.name
assert(mx_basis == 'pp' or mx_basis == 'gm' or mx_basis == 'std'), "mx_basis must be 'gm', 'pp' or 'std'."
for gate in list(target_model.operations.keys()):
if norm == 'diamond':
print(error_gs.operations[gate])
print(error_gs.operations['Gavg'])
delta.append(_optls.diamonddist(error_gs.operations[gate], error_gs.operations['Gavg'],
mx_basis=mx_basis))
elif norm == '1to1':
gate_dif = error_gs.operations[gate] - error_gs.operations['Gavg']
delta.append(_optls.norm1to1(gate_dif, num_samples=1000, mx_basis=mx_basis, return_list=False))
else:
raise ValueError("Only diamond or 1to1 norm available.")
delta_avg = _np.mean(delta)
return delta_avg
# Future : perhaps put these back in.
#def Magesan_theory_predicted_decay(model, target_model, mlist, success_outcomelabel=('0',),
# norm='1to1', order='zeroth', return_all = False):
#
# assert(order == 'zeroth' or order == 'first')
#
# d = int(round(_np.sqrt(model.dim)))
# MTPs = {}
# MTPs['r'] = gateset_infidelity(model,target_model,itype='AGI')
# MTPs['p'] = _analysis.r_to_p(MTPs['r'],d,rtype='AGI')
# MTPs['delta'] = gate_dependence_of_errormaps(model, target_model, norm)
# error_gs = errormaps(model, target_model)
#
# R_list = []
# Q_list = []
# for gate in list(target_model.operations.keys()):
# R_list.append(_np.dot(_np.dot(error_gs.operations[gate],target_model.operations[gate]),
# _np.dot(error_gs.operations['Gavg'],_np.transpose(target_model.operations[gate]))))
# Q_list.append(_np.dot(target_model.operations[gate],
# _np.dot(error_gs.operations[gate],_np.transpose(target_model.operations[gate]))))
#
# error_gs.operations['GR'] = _np.mean(_np.array([ i for i in R_list]),axis=0)
# error_gs.operations['GQ'] = _np.mean(_np.array([ i for i in Q_list]),axis=0)
# error_gs.operations['GQ2'] = _np.dot(error_gs.operations['GQ'],error_gs.operations['Gavg'])
# error_gs.preps['rhoc_mixed'] = 1./d*_cnst._basis_create_identity_vec(error_gs.basis)#
#
# #Assumes standard POVM labels
# povm = _objs.UnconstrainedPOVM( [('0_cm', target_model.povms['Mdefault']['0']),
# ('1_cm', target_model.povms['Mdefault']['1'])] )
# ave_error_gsl = _cnst.to_circuits([('rho0','Gavg'),('rho0','GR'),('rho0','Gavg','GQ')])
# data = _cnst.simulate_data(error_gs, ave_error_gsl, num_samples=1, sample_error="none")#
# pr_L_p = data[('rho0','Gavg')][success_outcomelabel]
# pr_L_I = data[('rho0','Gavg')][success_outcomelabel_cm]
# pr_R_p = data[('rho0','GR')][success_outcomelabel]
# pr_R_I = data[('rho0','GR')][success_outcomelabel_cm]
# pr_Q_p = data[('rho0','Gavg','GQ')][success_outcomelabel]
# p = MTPs['p']
# B_1 = pr_R_I
# A_1 = (pr_Q_p/p) - pr_L_p + ((p -1)*pr_L_I/p) + ((pr_R_p - pr_R_I)/p)
# C_1 = pr_L_p - pr_L_I
# q = _tls.average_gate_infidelity(error_gs.operations['GQ2'],_np.identity(d**2,float))
# q = _analysis.r_to_p(q,d,rtype='AGI')
#
# if order == 'zeroth':
# MTPs['A'] = pr_L_I
# MTPs['B'] = pr_L_p - pr_L_I
# if order == 'first':
# MTPs['A'] = B_1
# MTPs['B'] = A_1 - C_1*(q - 1)/p**2
# MTPs['C'] = C_1*(q- p**2)/p**2
#
# if order == 'zeroth':
# Pm = MTPs['A'] + MTPs['B']*MTPs['p']**_np.array(mlist)
# if order == 'first':
# Pm = MTPs['A'] + (MTPs['B'] + _np.array(mlist)*MTPs['C'])*MTPs['p']**_np.array(mlist)
#
# sys_eb = (MTPs['delta'] + 1)**(_np.array(mlist)+1) - 1
# if order == 'first':
# sys_eb = sys_eb - (_np.array(mlist)+1)*MTPs['delta']
#
# upper = Pm + sys_eb
# upper[upper > 1]=1.
#
# lower = Pm - sys_eb
# lower[lower < 0]=0.
#
# return mlist, Pm, upper, lower, MTPs