6391:
6381:
2450:
5369:. To recap, the quantum probability of a language being accepted can be interpreted as a metric, with the probability of accept being unity, if the metric distance between the initial and final states is zero, and otherwise the probability of accept is less than one, if the metric distance is non-zero. Thus, it follows that the quantum finite automaton is just a special case of a
897:; thus, this is a topological automaton, with the simplex being the manifold, and the stochastic matrices being linear automorphisms of the simplex onto itself. Since each transition is (essentially) independent of the previous (if we disregard the distinction between accepted and rejected languages), the PFA essentially becomes a kind of
893:. The entries in the state vector must be real numbers, positive, and sum to one, in order for the state vector to be interpreted as a probability. The transition matrices must preserve this property: this is why they must be stochastic. Each state vector should be imagined as specifying a point in a
310:
One needs a distinct adjacency matrix for each possible input symbol, since each input symbol can result in a different transition. The entries in the adjacency matrix must be zero's and one's. For any given column in the matrix, only one entry can be non-zero: this is the entry that indicates the
1025:
A worthy point to contemplate is the distributions that result on the manifold during the input of a language. In order for an automaton to be 'efficient' in recognizing a language, that distribution should be 'as uniform as possible'. This need for uniformity is the underlying principle behind
3452:
2811:
295:, with states as nodes in the graph, and arrows representing state transitions. Each arrow is labelled with a possible input symbol, so that, given a specific state and an input symbol, the arrow points at the next state. One way of representing such a graph is by means of a set of
5212:
The quantum automaton differs from the topological automaton in that, instead of having a binary result (is the iterated point in, or not in, the final set?), one has a probability. The quantum probability is the (square of) the initial state projected onto some final state
854:
are defined in such a way that a given column can have several non-zero entries in it. Equivalently, the multiply-add operations performed during component-wise matrix multiplication should be replaced by
Boolean and-or operations, that is, so that one is working with a
2016:
3122:, then the result of running the machine will be exactly identical to the classical deterministic finite state machine. In particular, there is a language accepted by this automaton with probability one, for these initial states, and it is identical to the
4241:
2595:
3659:
4706:
4249:
If the wave function has collapsed to either the "accept" or "reject" subspaces, then further processing halts. Otherwise, processing continues, with the next letter read from the input, and applied to what must be an eigenstate of
1613:
3324:
Measure-many automata were introduced by
Kondacs and Watrous in 1997. The general framework resembles that of the measure-once automaton, except that instead of there being one projection, at the end, there is a projection, or
3815:
2132:
1823:
5271:
5192:
appropriate for the given topological space. The initial state may be taken to be a point in the space. The set of accept states can be taken to be some arbitrary subset of the topological space. One then says that a
4375:
3375:
5037:
2649:
1897:
157:
2955:
2883:
3928:
1527:
3047:
4573:
637:
5108:
As a result of these effects, the actual time evolution of the state cannot be taken as an infinite-precision pure point, operated on by a sequence of arbitrarily sharp transformations, but rather as an
1905:
1217:
4880:
4152:
4024:
2350:
429:
4121:
4090:
3993:
3962:
311:
next (unique) state transition. Similarly, the state of the system is a column vector, in which only one entry is non-zero: this entry corresponds to the current state of the system. Let
3199:
242:
568:
3572:
3539:
4160:
4059:
2467:
1465:
3580:
4810:
The primary difference between real-world quantum computers and the theoretical framework presented above is that the initial state preparation cannot ever result in a point-like
5327:
3506:
3364:
1382:
1282:
355:
5407:
5359:
5182:
1020:
969:
937:
5301:
852:
768:
727:
5076:
4940:
4746:
4504:
4469:
4442:
4275:
3846:
3751:
3724:
3697:
3120:
3085:
2292:
2193:
2047:
1748:
1720:
1430:
1145:
4584:
2067:
303:. For a given input symbol, the adjacency matrix indicates how any given state (row in the state vector) will transition to the next state; a state transition is given by
5103:
4793:
2641:
2165:
1356:
675:
468:
382:
184:
509:
1650:
1327:
2049:
is understood to represent the initial state of the automaton, that is, the state the automaton was in before it started accepting the string input. The empty string
4524:
4415:
4295:
3866:
2312:
2244:
2213:
1402:
329:
207:
5580:
3310:
3283:
3256:
3229:
2985:
1249:
4912:
878:. In the generalization to QFAs, the set of recognized languages will be different. Describing that set is one of the outstanding research problems in QFA theory.
1049:
Although the study of QFA was popularized in the work of
Kondacs and Watrous in 1997 and later by Moore and Crutchfeld, they were described as early as 1971, by
5542:
4766:
4395:
3475:
2264:
1692:
1672:
1165:
1114:
1094:
1535:
5461:
6272:
6173:
5834:
3759:
797:
Before moving on to the formal description of a QFA, there are two noteworthy generalizations that should be mentioned and understood. The first is the
5201:
if the point, after iteration by the homeomorphisms, intersects the accept set. But, of course, this is nothing more than the standard definition of an
2075:
5735:
4950:
to describe the measurement process. Finally, each unitary transformation is not a single, sharply defined quantum logic gate, but rather a mixture
1756:
6060:
5220:
3447:{\displaystyle {\mathcal {H}}_{Q}={\mathcal {H}}_{\text{accept}}\oplus {\mathcal {H}}_{\text{reject}}\oplus {\mathcal {H}}_{\text{non-halting}}}
5480:
2806:{\displaystyle {\begin{bmatrix}a_{1}^{*}\;\;a_{2}^{*}\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}=a_{1}^{*}a_{1}+a_{2}^{*}a_{2}=1}
4303:
4956:
6384:
5570:
4720:
are implementations of measure-once quantum finite automata, and the software systems for programming them expose the state-preparation of
1831:
91:
2894:
2822:
6415:
6342:
5498:
3874:
1473:
5132:: there is no way to make a copy of the current state of the machine, push it onto a stack for later reference, and then return to it.
5918:
2993:
798:
4532:
573:
6394:
6282:
5535:
5456:
867:
5868:
5427:
6210:
2011:{\displaystyle \operatorname {Pr} (\sigma )=\Vert PU_{\sigma _{k}}\cdots U_{\sigma _{1}}U_{\sigma _{0}}|\psi \rangle \Vert ^{2}}
6205:
5933:
5913:
5712:
1043:
6200:
2361:
890:
805:
is replaced by a vector that can have more than one entry that is non-zero. Such a vector then represents an element of the
790:
of the manifold; this defines a topological finite automaton. Similarly, the matrices could be taken as automorphisms of a
3668:
of the basis vectors in the accept set. The reject space is defined analogously, and the remaining space is designated the
1170:
6233:
6055:
5958:
5619:
288:
268:
260:
4823:
4126:
3998:
6238:
6106:
5697:
5528:
2317:
387:
6018:
5878:
5652:
6262:
5607:
5551:
187:
5707:
4095:
4064:
3967:
3936:
6134:
6006:
5903:
5779:
5614:
5514:
I. Baianu, "Categories, Functors and
Quantum Automata Theory" (1971). The 4th Intl. Congress LMPS, August-Sept.1971
3136:
215:
5943:
5908:
5804:
5747:
4236:{\displaystyle \operatorname {Pr} _{\text{acc}}(\sigma )=\Vert P_{\text{acc}}|\psi ^{\prime }\rangle \Vert ^{2},}
2590:{\displaystyle |\psi \rangle =a_{1}|S_{1}\rangle +a_{2}|S_{2}\rangle ={\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}}}
544:
86:
6028:
5642:
4914:
characterizing the ability of the machinery to prepare an initial state close to the desired initial pure state
6420:
6116:
6089:
6065:
5819:
5752:
5687:
5672:
5140:
The above constructions indicate how the concept of a quantum finite automaton can be generalized to arbitrary
3654:{\displaystyle {\mathcal {H}}_{\text{accept}}=\operatorname {span} \{|q\rangle :|q\rangle \in Q_{\text{acc}}\}}
3544:
3511:
1252:
987:
905:
779:
67:
5565:
5366:
4029:
1435:
1285:
6267:
6001:
5893:
5863:
5662:
5149:
51:
6337:
6101:
6094:
5841:
2365:
1027:
976:
47:
5306:
3316:
as an example of a quantum finite state machine acting on the set of all possible finite binary strings.
6257:
5809:
5774:
5274:
3820:
and so on. The parsing of the input string proceeds as follows. Consider the automaton to be in a state
3480:
3338:
1616:
1361:
1261:
980:
640:
334:
304:
299:, with one matrix for each input symbol. In this case, the list of possible DFA states is written as a
5883:
5380:
5332:
5155:
4277:. Processing continues until the whole string is read, or the machine halts. Often, additional symbols
2459:
993:
942:
910:
4701:{\displaystyle U_{\alpha }|q_{i}\rangle =\sum _{q_{j}\in Q}\delta (q_{i},\alpha ,q_{j})|q_{j}\rangle }
6047:
5796:
5647:
5422:
5362:
5280:
5206:
1066:
824:
740:
699:
6149:
5045:
4917:
4723:
4474:
4447:
4420:
4253:
3823:
3729:
3702:
3675:
3090:
3055:
2269:
2170:
2024:
1725:
1697:
1407:
1122:
6366:
6319:
5923:
5677:
5657:
5592:
5587:
5121:
4943:
4815:
4796:
3367:
3326:
2216:
1035:
975:; the unitary matrices can be thought of as governing the time evolution of the system (viz in the
264:
4814:, nor can the unitary operators be precisely applied. Thus, the initial state must be taken as a
2052:
6082:
5730:
5667:
5129:
5081:
4804:
4800:
4771:
3127:
2606:
2143:
1334:
886:
856:
814:
653:
446:
360:
162:
5928:
1050:
481:
1629:
1306:
6346:
5991:
5898:
5855:
5786:
5702:
5682:
5637:
5597:
5575:
5141:
4297:
and $ are adjoined to the alphabet, to act as the left and right end-markers for the string.
2449:
1652:
1296:
1039:
882:
791:
527:
31:
4509:
4400:
4280:
3851:
2297:
2229:
2198:
1387:
314:
283:
There is a simple, intuitive way of understanding quantum finite automata. One begins with a
192:
6013:
5963:
5740:
5413:, and the probability measure is replaced by a simple function of the metric on that space.
5114:
4717:
3123:
1329:
1292:
1073:
1031:
1030:: these simply guarantee crisp, compact operation of the automaton. Put in other words, the
875:
860:
384:
as the adjacency matrix that describes the evolution of the DFA to its next state. The set
296:
256:
55:
3288:
3261:
3234:
3207:
2963:
1222:
6139:
6077:
5767:
5762:
5194:
5117:
that only concatenates transformations onto a state, but also smears the state over time.
4888:
2223:
1300:
871:
689:
681:); the order of application is 'reversed' only because we follow the standard notation of
252:
3457:
In the literature, these orthogonal subspaces are usually formulated in terms of the set
1608:{\displaystyle (P(\mathbb {C} ^{N}),\Sigma ,\{U_{\alpha }\;\vert \;\alpha \in \Sigma \})}
6248:
6225:
6192:
5996:
5873:
4751:
4380:
3460:
2601:
2249:
1677:
1657:
1150:
1099:
1079:
984:
771:
682:
292:
17:
1404:, the unitary matrix describes the transition of the automaton from its current state
6409:
6070:
5888:
5814:
5432:
5202:
5189:
5125:
3333:
3313:
1256:
972:
787:
300:
6290:
6215:
5410:
3810:{\displaystyle P_{\text{acc}}:{\mathcal {H}}_{Q}\to {\mathcal {H}}_{\text{accept}}}
1623:
898:
284:
245:
71:
2127:{\displaystyle \operatorname {Pr} (\varnothing )=\Vert P|\psi \rangle \Vert ^{2}}
66:
automata. Quantum finite automata can also be understood as the quantization of
6300:
6154:
5692:
3665:
866:
A well-known theorem states that, for each DFA, there is an equivalent NFA, and
734:
210:
6361:
6295:
6159:
5520:
4811:
1062:
1818:{\displaystyle \langle \psi |P|\psi \rangle =\Vert P|\psi \rangle \Vert ^{2}}
248:; that is, indicating whether the automaton accepted or rejected the string.
6144:
4946:
over time. Precise measurements are also not possible, and one instead uses
4300:
In the literature, the measure-many automaton is often denoted by the tuple
4061:, at which time its wave-function collapses into one of the three subspaces
3933:
At this point, a measurement whose three possible outcomes have eigenspaces
806:
729:
by some general operators. This is essentially what a QFA does: it replaces
5462:
Proceedings of the 38th Annual
Symposium on Foundations of Computer Science
1828:
The probability of the state machine accepting a given finite input string
692:
and vectors, almost begs for generalization, by replacing the state-vector
5266:{\displaystyle \mathbf {Pr} =\vert \langle P\vert \psi \rangle \vert ^{2}}
6329:
6305:
6164:
6129:
5185:
783:
5078:
describing how well the machinery can effect the desired transformation
2215:, it is not uncommon for QFAs to be defined using a right action on the
431:
then completely describes the state transition function of the DFA. Let
6356:
5973:
5110:
4370:{\displaystyle (Q;\Sigma ;\delta ;q_{0};Q_{\text{acc}};Q_{\text{rej}})}
894:
881:
Another generalization that should be immediately apparent is to use a
5032:{\displaystyle U_{\alpha ,(\rho )}=\int p_{\alpha }(x)U_{\alpha ,x}dx}
3052:
As should be readily apparent, if the initial state is the pure state
2137:
is just the probability of the initial state being an accepted state.
874:
that can be recognized by DFA's and NFA's are the same; these are the
6333:
5829:
1892:{\displaystyle \sigma =(\sigma _{0},\sigma _{1},\cdots ,\sigma _{k})}
152:{\displaystyle \sigma =(\sigma _{0},\sigma _{1},\cdots ,\sigma _{k})}
3329:, performed after each letter is read. A formal definition follows.
2950:{\displaystyle U_{1}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}}
2878:{\displaystyle U_{0}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}}
5478:
C. Moore, J. Crutchfield, "Quantum automata and quantum grammars",
5205:. The behaviour of topological automata is studied in the field of
4154:. The probability of collapse to the "accept" subspace is given by
2219:
states, simply in order to keep the order of the letters the same.
939:, and the transition matrices are unitary matrices. Each point in
5602:
3923:{\displaystyle |\psi ^{\prime }\rangle =U_{\alpha }|\psi \rangle }
1522:{\displaystyle |\psi ^{\prime }\rangle =U_{\alpha }|\psi \rangle }
1117:
774:. Other, similar generalizations also become obvious: the vector
5277:
is just a very simple function of the distance between the point
541:
also denote these two vectors. Then, after reading input symbols
6351:
5824:
5757:
4947:
3042:{\displaystyle P={\begin{bmatrix}1&0\\0&0\end{bmatrix}}}
2266:
by a quantum finite automaton (and a given, fixed initial state
5524:
4568:{\displaystyle \delta :Q\times \Sigma \times Q\to \mathbb {C} }
632:{\displaystyle q=\cdots U_{\gamma }U_{\beta }U_{\alpha }q_{0}.}
5968:
5953:
5499:
435:
represent the set of possible states of the DFA. If there are
3258:
are non-zero. More subtle behaviour occurs when the matrices
5188:
of the
Riemannian manifold, or, more generally, some set of
4942:. This state is not stable, but suffers from some amount of
4212:
4133:
4102:
4071:
4043:
4005:
3974:
3943:
3888:
3796:
3779:
3587:
3487:
3433:
3416:
3399:
3382:
3345:
1487:
1449:
2373:
570:
from the input tape, the state of the DFA will be given by
331:
denote the set of input symbols. For a given input symbol
5459:(1997), "On the power of quantum finite state automata",
1096:
possible internal states, represented in this case by an
244:
indicating the probability of the automaton being in an
54:. They provide a mathematical abstraction of real-world
3508:. This set of basis vectors is divided up into subsets
4471:
are as defined above. The initial state is denoted by
3008:
2916:
2844:
2706:
2658:
2552:
58:. Several types of automata may be defined, including
5383:
5335:
5309:
5283:
5223:
5158:
5084:
5048:
4959:
4920:
4891:
4826:
4774:
4754:
4726:
4587:
4535:
4512:
4506:. The unitary transformations are denoted by the map
4477:
4450:
4423:
4403:
4383:
4306:
4283:
4256:
4163:
4129:
4098:
4067:
4032:
4001:
3970:
3939:
3877:
3854:
3826:
3762:
3732:
3705:
3678:
3583:
3547:
3514:
3483:
3463:
3378:
3341:
3291:
3264:
3237:
3210:
3139:
3093:
3058:
2996:
2966:
2897:
2825:
2652:
2609:
2470:
2320:
2300:
2272:
2252:
2232:
2201:
2173:
2146:
2078:
2055:
2027:
1908:
1834:
1759:
1728:
1700:
1680:
1660:
1632:
1538:
1476:
1438:
1410:
1390:
1364:
1337:
1309:
1264:
1225:
1212:{\displaystyle |\psi \rangle \in P(\mathbb {C} ^{N})}
1173:
1153:
1125:
1102:
1082:
996:
983:
should be straightforward: A mixed state is simply a
945:
913:
827:
743:
702:
656:
576:
547:
484:
449:
390:
363:
337:
317:
218:
195:
165:
94:
6318:
6281:
6247:
6224:
6191:
6182:
6115:
6044:
5982:
5942:
5854:
5795:
5721:
5630:
5558:
4875:{\displaystyle \rho =\int p(x)|\psi _{x}\rangle dx}
4147:{\displaystyle {\mathcal {H}}_{\text{non-halting}}}
4019:{\displaystyle {\mathcal {H}}_{\text{non-halting}}}
5401:
5353:
5321:
5295:
5265:
5176:
5097:
5070:
5031:
4934:
4906:
4874:
4787:
4760:
4740:
4700:
4567:
4518:
4498:
4463:
4436:
4409:
4389:
4369:
4289:
4269:
4235:
4146:
4115:
4084:
4053:
4018:
3987:
3956:
3922:
3860:
3840:
3809:
3745:
3718:
3691:
3653:
3566:
3533:
3500:
3477:of orthogonal basis vectors for the Hilbert space
3469:
3446:
3358:
3304:
3277:
3250:
3223:
3193:
3114:
3079:
3041:
2979:
2949:
2877:
2805:
2635:
2589:
2345:{\displaystyle p\leq \operatorname {Pr} (\sigma )}
2344:
2306:
2286:
2258:
2238:
2207:
2187:
2159:
2126:
2069:is understood to be just the unit matrix, so that
2061:
2041:
2010:
1891:
1817:
1742:
1714:
1686:
1666:
1644:
1607:
1521:
1459:
1424:
1396:
1376:
1350:
1321:
1276:
1243:
1211:
1159:
1139:
1108:
1088:
1014:
963:
931:
846:
762:
721:
669:
631:
562:
503:
462:
424:{\displaystyle \{U_{\alpha }|\alpha \in \Sigma \}}
423:
376:
349:
323:
236:
201:
178:
151:
5184:. In place of the unitary matrices, one uses the
3190:
2987:to be the accept state, the projection matrix is
511:corresponds to a column vector with a one in the
3672:subspace. There are three projection matrices,
2195:reverses the order of the letters in the string
5501:" (1971), Bulletin of Mathematical Biophysics,
85:The automata work by receiving a finite-length
4116:{\displaystyle {\mathcal {H}}_{\text{reject}}}
4085:{\displaystyle {\mathcal {H}}_{\text{accept}}}
3988:{\displaystyle {\mathcal {H}}_{\text{reject}}}
3957:{\displaystyle {\mathcal {H}}_{\text{accept}}}
3753:, each projecting to the respective subspace:
1076:, the quantum automaton is considered to have
5536:
794:; this defines a geometric finite automaton.
8:
5316:
5310:
5290:
5284:
5254:
5250:
5244:
5238:
5235:
4929:
4863:
4735:
4695:
4613:
4493:
4221:
4217:
4189:
4048:
3917:
3893:
3835:
3648:
3632:
3618:
3607:
3194:{\displaystyle (1^{*}(01^{*}0)^{*})^{*}\,\!}
3109:
3074:
2541:
2510:
2479:
2281:
2182:
2115:
2111:
2097:
2036:
1999:
1995:
1927:
1806:
1802:
1788:
1782:
1760:
1737:
1709:
1599:
1586:
1572:
1516:
1492:
1454:
1419:
1271:
1265:
1182:
1134:
841:
828:
757:
744:
716:
703:
688:The above description of a DFA, in terms of
639:The state transitions are given by ordinary
418:
391:
237:{\displaystyle \operatorname {Pr} (\sigma )}
3204:The non-classical behaviour occurs if both
3126:for the classical DFA, and is given by the
1069:. They may be defined formally as follows.
563:{\displaystyle \alpha \beta \gamma \cdots }
6188:
5792:
5543:
5529:
5521:
4246:and analogously for the other two spaces.
2677:
2676:
1589:
1585:
1358:, with one unitary matrix for each letter
979:). The generalization from pure states to
821:. Likewise, the state transition matrices
522:is then a column vector with a one in the
5450:
5448:
5393:
5385:
5384:
5382:
5345:
5337:
5336:
5334:
5308:
5282:
5257:
5224:
5222:
5168:
5160:
5159:
5157:
5089:
5083:
5053:
5047:
5011:
4992:
4964:
4958:
4921:
4919:
4890:
4857:
4848:
4825:
4779:
4773:
4753:
4727:
4725:
4689:
4680:
4671:
4652:
4628:
4623:
4607:
4598:
4592:
4586:
4561:
4560:
4534:
4511:
4487:
4478:
4476:
4455:
4449:
4428:
4422:
4402:
4382:
4358:
4345:
4332:
4305:
4282:
4261:
4255:
4224:
4211:
4202:
4196:
4168:
4162:
4138:
4132:
4131:
4128:
4107:
4101:
4100:
4097:
4076:
4070:
4069:
4066:
4042:
4033:
4031:
4010:
4004:
4003:
4000:
3979:
3973:
3972:
3969:
3948:
3942:
3941:
3938:
3909:
3903:
3887:
3878:
3876:
3853:
3827:
3825:
3801:
3795:
3794:
3784:
3778:
3777:
3767:
3761:
3737:
3731:
3710:
3704:
3683:
3677:
3642:
3624:
3610:
3592:
3586:
3585:
3582:
3567:{\displaystyle Q_{\text{rej}}\subseteq Q}
3552:
3546:
3534:{\displaystyle Q_{\text{acc}}\subseteq Q}
3519:
3513:
3492:
3486:
3485:
3482:
3462:
3438:
3432:
3431:
3421:
3415:
3414:
3404:
3398:
3397:
3387:
3381:
3380:
3377:
3350:
3344:
3343:
3340:
3312:are not so simple; see, for example, the
3296:
3290:
3269:
3263:
3242:
3236:
3215:
3209:
3189:
3183:
3173:
3160:
3147:
3138:
3103:
3094:
3092:
3068:
3059:
3057:
3003:
2995:
2971:
2965:
2911:
2902:
2896:
2839:
2830:
2824:
2791:
2781:
2776:
2763:
2753:
2748:
2727:
2713:
2701:
2687:
2682:
2670:
2665:
2653:
2651:
2627:
2614:
2608:
2573:
2559:
2547:
2535:
2526:
2520:
2504:
2495:
2489:
2471:
2469:
2391:
2386:
2319:
2299:
2273:
2271:
2251:
2231:
2200:
2174:
2172:
2151:
2145:
2118:
2103:
2077:
2054:
2028:
2026:
2002:
1987:
1979:
1974:
1962:
1957:
1942:
1937:
1907:
1880:
1861:
1848:
1833:
1809:
1794:
1774:
1766:
1758:
1729:
1727:
1701:
1699:
1679:
1659:
1631:
1579:
1554:
1550:
1549:
1537:
1508:
1502:
1486:
1477:
1475:
1448:
1439:
1437:
1411:
1409:
1389:
1363:
1342:
1336:
1308:
1263:
1224:
1200:
1196:
1195:
1174:
1172:
1152:
1126:
1124:
1101:
1081:
1061:Measure-once automata were introduced by
1006:
998:
997:
995:
955:
947:
946:
944:
923:
915:
914:
912:
835:
826:
751:
742:
710:
701:
696:by some general vector, and the matrices
661:
655:
620:
610:
600:
590:
575:
546:
489:
483:
454:
448:
404:
398:
389:
368:
362:
336:
316:
217:
194:
170:
164:
140:
121:
108:
93:
4768:and a choice of unitary transformations
4054:{\displaystyle |\psi ^{\prime }\rangle }
1460:{\displaystyle |\psi ^{\prime }\rangle }
786:; the set of transition matrices become
27:Quantum analog of probabilistic automata
6061:Continuous-variable quantum information
5474:
5472:
5444:
2088:
2056:
904:By contrast, in a QFA, the manifold is
291:(DFA). A DFA can be represented as a
209:, and assigning to each such string a
74:. QFAs are, in turn, special cases of
3868:, the automaton will be in the state
7:
2816:The unitary transition matrices are
275:remains an active area of research.
5322:{\displaystyle \vert \psi \rangle }
1303:are represented by a collection of
885:for the transition matrices, and a
5144:. For example, one may take some (
5120:There is no quantum analog to the
5042:for some probability distribution
4885:for some probability distribution
4548:
4404:
4316:
3501:{\displaystyle {\mathcal {H}}_{Q}}
3359:{\displaystyle {\mathcal {H}}_{Q}}
2458:The quantum state is a vector, in
2233:
1596:
1566:
1377:{\displaystyle \alpha \in \Sigma }
1371:
1277:{\displaystyle \Vert \cdot \Vert }
799:non-deterministic finite automaton
415:
350:{\displaystyle \alpha \in \Sigma }
344:
318:
196:
25:
5402:{\displaystyle \mathbb {C} P^{N}}
5354:{\displaystyle \mathbb {C} P^{N}}
5177:{\displaystyle \mathbb {C} P^{N}}
4948:positive operator-valued measures
1384:. That is, given an input letter
1015:{\displaystyle \mathbb {C} P^{N}}
964:{\displaystyle \mathbb {C} P^{N}}
932:{\displaystyle \mathbb {C} P^{N}}
801:(NFA). In this case, the vector
6390:
6389:
6380:
6379:
5228:
5225:
3848:. After reading an input letter
2448:
1626:of the automaton is given by an
1038:generalize to QFAs as well: the
870:. This implies that the set of
478:-dimensional. The initial state
5296:{\displaystyle \vert P\rangle }
5113:process, or more accurately, a
1046:generalize readily to the QFA.
847:{\displaystyle \{U_{\alpha }\}}
763:{\displaystyle \{U_{\alpha }\}}
722:{\displaystyle \{U_{\alpha }\}}
5071:{\displaystyle p_{\alpha }(x)}
5065:
5059:
5004:
4998:
4977:
4971:
4935:{\displaystyle |\psi \rangle }
4922:
4901:
4895:
4849:
4845:
4839:
4807:, directly to the programmer.
4741:{\displaystyle |\psi \rangle }
4728:
4681:
4677:
4645:
4599:
4557:
4499:{\displaystyle |q_{0}\rangle }
4479:
4464:{\displaystyle Q_{\text{rej}}}
4437:{\displaystyle Q_{\text{acc}}}
4364:
4307:
4270:{\displaystyle P_{\text{non}}}
4203:
4183:
4177:
4034:
3910:
3879:
3841:{\displaystyle |\psi \rangle }
3828:
3790:
3746:{\displaystyle P_{\text{non}}}
3719:{\displaystyle P_{\text{rej}}}
3692:{\displaystyle P_{\text{acc}}}
3625:
3611:
3180:
3170:
3153:
3140:
3115:{\displaystyle |S_{2}\rangle }
3095:
3080:{\displaystyle |S_{1}\rangle }
3060:
2527:
2496:
2472:
2362:deterministic finite automaton
2339:
2333:
2287:{\displaystyle |\psi \rangle }
2274:
2188:{\displaystyle |\psi \rangle }
2175:
2104:
2091:
2085:
2042:{\displaystyle |\psi \rangle }
2029:
1988:
1921:
1915:
1886:
1841:
1795:
1775:
1767:
1743:{\displaystyle |\psi \rangle }
1730:
1715:{\displaystyle |\psi \rangle }
1702:
1602:
1560:
1545:
1539:
1509:
1478:
1440:
1425:{\displaystyle |\psi \rangle }
1412:
1238:
1226:
1206:
1191:
1175:
1140:{\displaystyle |\psi \rangle }
1127:
891:probabilistic finite automaton
405:
231:
225:
146:
101:
1:
6056:Adiabatic quantum computation
4712:Relation to quantum computing
2246:is accepted with probability
1750:being in the accept state is
289:deterministic finite automata
269:probabilistic finite automata
261:deterministic finite automata
255:accepted by QFAs are not the
6107:Topological quantum computer
5481:Theoretical Computer Science
2443:
2430:
2424:
2410:
2404:
2062:{\displaystyle \varnothing }
889:for the state; this gives a
6385:Quantum information science
5552:Quantum information science
5098:{\displaystyle U_{\alpha }}
4788:{\displaystyle U_{\alpha }}
2636:{\displaystyle a_{1},a_{2}}
2418:
2398:
2160:{\displaystyle U_{\alpha }}
2140:Because the left-action of
1694:-dimensional quantum state
1351:{\displaystyle U_{\alpha }}
670:{\displaystyle U_{\alpha }}
463:{\displaystyle U_{\alpha }}
377:{\displaystyle U_{\alpha }}
179:{\displaystyle \sigma _{i}}
80:topological finite automata
6437:
6416:Quantum information theory
5780:quantum gate teleportation
4026:is performed on the state
1044:forward–backward algorithm
504:{\displaystyle q_{0}\in Q}
70:, or as a quantization of
6375:
5909:Quantum Fourier transform
5805:Post-quantum cryptography
5748:Entanglement distillation
5136:Geometric generalizations
3366:is decomposed into three
2314:in the language, one has
2294:), if, for all sentences
1645:{\displaystyle N\times N}
1322:{\displaystyle N\times N}
518:'th row. A general state
76:geometric finite automata
6395:Quantum mechanics topics
6090:Quantum machine learning
6066:One-way quantum computer
5919:Quantum phase estimation
5820:Quantum key distribution
5753:Monogamy of entanglement
1253:complex projective space
988:probability distribution
973:quantum-mechanical state
971:corresponds to a (pure)
906:complex projective space
68:subshifts of finite type
46:are a quantum analog of
6002:Randomized benchmarking
5864:Amplitude amplification
5428:Blum–Shub–Smale machine
5409:is generalized to some
5150:Riemann symmetric space
4519:{\displaystyle \delta }
4410:{\displaystyle \Sigma }
4290:{\displaystyle \kappa }
3861:{\displaystyle \alpha }
2360:Consider the classical
2307:{\displaystyle \sigma }
2239:{\displaystyle \Sigma }
2208:{\displaystyle \sigma }
1397:{\displaystyle \alpha }
1028:maximum entropy methods
324:{\displaystyle \Sigma }
202:{\displaystyle \Sigma }
52:Markov decision process
36:quantum finite automata
18:Quantum finite automata
6102:Quantum Turing machine
6095:quantum neural network
5842:Quantum secret sharing
5403:
5355:
5323:
5297:
5267:
5178:
5099:
5072:
5033:
4936:
4908:
4876:
4789:
4762:
4742:
4702:
4569:
4520:
4500:
4465:
4438:
4411:
4391:
4371:
4291:
4271:
4237:
4148:
4117:
4086:
4055:
4020:
3989:
3958:
3924:
3862:
3842:
3811:
3747:
3720:
3693:
3655:
3568:
3535:
3502:
3471:
3448:
3360:
3306:
3279:
3252:
3225:
3195:
3116:
3081:
3043:
2981:
2951:
2879:
2807:
2637:
2591:
2376:State Transition Table
2366:state transition table
2346:
2308:
2288:
2260:
2240:
2209:
2189:
2161:
2128:
2063:
2043:
2012:
1893:
1819:
1744:
1716:
1688:
1668:
1646:
1609:
1523:
1461:
1426:
1398:
1378:
1352:
1323:
1278:
1245:
1213:
1161:
1147:. More precisely, the
1141:
1110:
1090:
1034:methods used to train
1016:
965:
933:
848:
764:
723:
671:
633:
564:
505:
464:
425:
378:
351:
325:
238:
203:
180:
153:
48:probabilistic automata
44:quantum state machines
6174:Entanglement-assisted
6135:quantum convolutional
5810:Quantum coin flipping
5775:Quantum teleportation
5736:entanglement-assisted
5566:DiVincenzo's criteria
5404:
5361:, under the distance
5356:
5324:
5298:
5275:probability amplitude
5268:
5199:topological automaton
5179:
5152:to take the place of
5128:. This is due to the
5100:
5073:
5034:
4937:
4909:
4877:
4790:
4763:
4743:
4703:
4570:
4521:
4501:
4466:
4439:
4412:
4392:
4372:
4292:
4272:
4238:
4149:
4118:
4087:
4056:
4021:
3990:
3959:
3925:
3863:
3843:
3812:
3748:
3721:
3694:
3656:
3569:
3536:
3503:
3472:
3449:
3361:
3320:Measure-many automata
3307:
3305:{\displaystyle U_{1}}
3280:
3278:{\displaystyle U_{0}}
3253:
3251:{\displaystyle a_{2}}
3226:
3224:{\displaystyle a_{1}}
3196:
3117:
3082:
3044:
2982:
2980:{\displaystyle S_{1}}
2952:
2880:
2808:
2638:
2592:
2347:
2309:
2289:
2261:
2241:
2210:
2190:
2162:
2129:
2064:
2044:
2013:
1894:
1820:
1745:
1722:, the probability of
1717:
1689:
1669:
1647:
1617:quantum semiautomaton
1610:
1524:
1462:
1427:
1399:
1379:
1353:
1324:
1279:
1246:
1244:{\displaystyle (N-1)}
1214:
1162:
1142:
1111:
1091:
1057:Measure-once automata
1017:
966:
934:
849:
765:
724:
672:
641:matrix multiplication
634:
565:
506:
465:
426:
379:
352:
326:
305:matrix multiplication
239:
204:
181:
154:
5985:processor benchmarks
5914:Quantum optimization
5797:Quantum cryptography
5608:physical vs. logical
5423:Quantum Markov chain
5381:
5333:
5307:
5281:
5221:
5207:topological dynamics
5197:is accepted by this
5156:
5082:
5046:
4957:
4918:
4907:{\displaystyle p(x)}
4889:
4824:
4772:
4752:
4724:
4585:
4533:
4510:
4475:
4448:
4421:
4401:
4381:
4304:
4281:
4254:
4161:
4127:
4096:
4065:
4030:
3999:
3968:
3937:
3875:
3852:
3824:
3760:
3730:
3703:
3676:
3581:
3545:
3512:
3481:
3461:
3376:
3368:orthogonal subspaces
3339:
3289:
3262:
3235:
3208:
3137:
3091:
3056:
2994:
2964:
2895:
2823:
2650:
2607:
2468:
2318:
2298:
2270:
2250:
2230:
2199:
2171:
2144:
2076:
2053:
2025:
1906:
1832:
1757:
1726:
1698:
1678:
1658:
1630:
1536:
1474:
1436:
1408:
1388:
1362:
1335:
1307:
1262:
1223:
1171:
1151:
1123:
1100:
1080:
1072:As with an ordinary
1067:James P. Crutchfield
1036:hidden Markov models
994:
943:
911:
825:
741:
700:
654:
574:
545:
482:
447:
388:
361:
335:
315:
279:Informal description
265:stochastic languages
216:
193:
163:
92:
5698:Quantum speed limit
5593:Quantum programming
5588:Quantum information
5371:geometric automaton
5367:Fubini–Study metric
5122:push-down automaton
4944:quantum decoherence
4805:quantum logic gates
4797:controlled NOT gate
3327:quantum measurement
2786:
2758:
2692:
2675:
2643:normalized so that
2378:
2217:Hermitian transpose
1674:, so that, given a
1297:transition matrices
1286:Fubini–Study metric
977:Schrödinger picture
643:(that is, multiply
443:, then each matrix
263:, nor are they the
6347:Forest/Rigetti QCS
6083:quantum logic gate
5869:Bernstein–Vazirani
5856:Quantum algorithms
5731:Classical capacity
5615:Quantum processors
5598:Quantum simulation
5488:(2000) pp 275-306.
5399:
5351:
5319:
5293:
5263:
5174:
5142:topological spaces
5130:no-cloning theorem
5095:
5068:
5029:
4932:
4904:
4872:
4801:Hadamard transform
4785:
4758:
4738:
4698:
4641:
4565:
4516:
4496:
4461:
4434:
4407:
4387:
4367:
4287:
4267:
4233:
4144:
4113:
4082:
4051:
4016:
3985:
3954:
3920:
3858:
3838:
3807:
3743:
3716:
3689:
3651:
3564:
3531:
3498:
3467:
3444:
3356:
3302:
3275:
3248:
3221:
3191:
3128:regular expression
3112:
3077:
3039:
3033:
2977:
2947:
2941:
2875:
2869:
2803:
2772:
2744:
2735:
2695:
2678:
2661:
2633:
2587:
2581:
2374:
2342:
2304:
2284:
2256:
2236:
2226:over the alphabet
2205:
2185:
2157:
2124:
2059:
2039:
2008:
1889:
1815:
1740:
1712:
1684:
1664:
1642:
1605:
1519:
1457:
1432:to its next state
1422:
1394:
1374:
1348:
1319:
1274:
1241:
1209:
1157:
1137:
1106:
1086:
1012:
961:
929:
887:probability vector
844:
815:indicator function
760:
719:
667:
629:
560:
501:
460:
421:
374:
347:
321:
297:adjacency matrices
287:interpretation of
234:
199:
176:
149:
6403:
6402:
6314:
6313:
6211:Linear optical QC
5992:Quantum supremacy
5946:complexity theory
5899:Quantum annealing
5850:
5849:
5787:Superdense coding
5576:Quantum computing
4761:{\displaystyle P}
4718:quantum computers
4716:As of 2019, most
4619:
4458:
4431:
4390:{\displaystyle Q}
4361:
4348:
4264:
4199:
4171:
4141:
4110:
4079:
4013:
3982:
3951:
3804:
3770:
3740:
3713:
3686:
3645:
3595:
3555:
3522:
3470:{\displaystyle Q}
3441:
3424:
3407:
2456:
2455:
2438:
2437:
2382: Input
2259:{\displaystyle p}
2021:Here, the vector
1687:{\displaystyle N}
1667:{\displaystyle P}
1653:projection matrix
1532:Thus, the triple
1293:state transitions
1219:is an element of
1160:{\displaystyle N}
1109:{\displaystyle N}
1089:{\displaystyle N}
1040:Viterbi algorithm
985:measure-theoretic
883:stochastic matrix
876:regular languages
792:homogeneous space
528:abuse of notation
273:quantum languages
271:. Study of these
257:regular languages
56:quantum computers
32:quantum computing
16:(Redirected from
6428:
6393:
6392:
6383:
6382:
6189:
6119:error correction
6048:computing models
6014:Relaxation times
5904:Quantum counting
5793:
5741:quantum capacity
5688:No-teleportation
5673:No-communication
5545:
5538:
5531:
5522:
5515:
5512:
5506:
5495:
5489:
5476:
5467:
5466:
5465:, pp. 66–75
5452:
5408:
5406:
5405:
5400:
5398:
5397:
5388:
5375:metric automaton
5360:
5358:
5357:
5352:
5350:
5349:
5340:
5328:
5326:
5325:
5320:
5302:
5300:
5299:
5294:
5272:
5270:
5269:
5264:
5262:
5261:
5231:
5183:
5181:
5180:
5175:
5173:
5172:
5163:
5104:
5102:
5101:
5096:
5094:
5093:
5077:
5075:
5074:
5069:
5058:
5057:
5038:
5036:
5035:
5030:
5022:
5021:
4997:
4996:
4981:
4980:
4941:
4939:
4938:
4933:
4925:
4913:
4911:
4910:
4905:
4881:
4879:
4878:
4873:
4862:
4861:
4852:
4794:
4792:
4791:
4786:
4784:
4783:
4767:
4765:
4764:
4759:
4747:
4745:
4744:
4739:
4731:
4707:
4705:
4704:
4699:
4694:
4693:
4684:
4676:
4675:
4657:
4656:
4640:
4633:
4632:
4612:
4611:
4602:
4597:
4596:
4574:
4572:
4571:
4566:
4564:
4525:
4523:
4522:
4517:
4505:
4503:
4502:
4497:
4492:
4491:
4482:
4470:
4468:
4467:
4462:
4460:
4459:
4456:
4443:
4441:
4440:
4435:
4433:
4432:
4429:
4416:
4414:
4413:
4408:
4396:
4394:
4393:
4388:
4376:
4374:
4373:
4368:
4363:
4362:
4359:
4350:
4349:
4346:
4337:
4336:
4296:
4294:
4293:
4288:
4276:
4274:
4273:
4268:
4266:
4265:
4262:
4242:
4240:
4239:
4234:
4229:
4228:
4216:
4215:
4206:
4201:
4200:
4197:
4173:
4172:
4169:
4153:
4151:
4150:
4145:
4143:
4142:
4139:
4137:
4136:
4122:
4120:
4119:
4114:
4112:
4111:
4108:
4106:
4105:
4091:
4089:
4088:
4083:
4081:
4080:
4077:
4075:
4074:
4060:
4058:
4057:
4052:
4047:
4046:
4037:
4025:
4023:
4022:
4017:
4015:
4014:
4011:
4009:
4008:
3994:
3992:
3991:
3986:
3984:
3983:
3980:
3978:
3977:
3963:
3961:
3960:
3955:
3953:
3952:
3949:
3947:
3946:
3929:
3927:
3926:
3921:
3913:
3908:
3907:
3892:
3891:
3882:
3867:
3865:
3864:
3859:
3847:
3845:
3844:
3839:
3831:
3816:
3814:
3813:
3808:
3806:
3805:
3802:
3800:
3799:
3789:
3788:
3783:
3782:
3772:
3771:
3768:
3752:
3750:
3749:
3744:
3742:
3741:
3738:
3725:
3723:
3722:
3717:
3715:
3714:
3711:
3698:
3696:
3695:
3690:
3688:
3687:
3684:
3660:
3658:
3657:
3652:
3647:
3646:
3643:
3628:
3614:
3597:
3596:
3593:
3591:
3590:
3573:
3571:
3570:
3565:
3557:
3556:
3553:
3540:
3538:
3537:
3532:
3524:
3523:
3520:
3507:
3505:
3504:
3499:
3497:
3496:
3491:
3490:
3476:
3474:
3473:
3468:
3453:
3451:
3450:
3445:
3443:
3442:
3439:
3437:
3436:
3426:
3425:
3422:
3420:
3419:
3409:
3408:
3405:
3403:
3402:
3392:
3391:
3386:
3385:
3365:
3363:
3362:
3357:
3355:
3354:
3349:
3348:
3311:
3309:
3308:
3303:
3301:
3300:
3284:
3282:
3281:
3276:
3274:
3273:
3257:
3255:
3254:
3249:
3247:
3246:
3230:
3228:
3227:
3222:
3220:
3219:
3200:
3198:
3197:
3192:
3188:
3187:
3178:
3177:
3165:
3164:
3152:
3151:
3124:regular language
3121:
3119:
3118:
3113:
3108:
3107:
3098:
3086:
3084:
3083:
3078:
3073:
3072:
3063:
3048:
3046:
3045:
3040:
3038:
3037:
2986:
2984:
2983:
2978:
2976:
2975:
2956:
2954:
2953:
2948:
2946:
2945:
2907:
2906:
2884:
2882:
2881:
2876:
2874:
2873:
2835:
2834:
2812:
2810:
2809:
2804:
2796:
2795:
2785:
2780:
2768:
2767:
2757:
2752:
2740:
2739:
2732:
2731:
2718:
2717:
2700:
2699:
2691:
2686:
2674:
2669:
2642:
2640:
2639:
2634:
2632:
2631:
2619:
2618:
2596:
2594:
2593:
2588:
2586:
2585:
2578:
2577:
2564:
2563:
2540:
2539:
2530:
2525:
2524:
2509:
2508:
2499:
2494:
2493:
2475:
2460:bra–ket notation
2452:
2379:
2370:
2369:
2351:
2349:
2348:
2343:
2313:
2311:
2310:
2305:
2293:
2291:
2290:
2285:
2277:
2265:
2263:
2262:
2257:
2245:
2243:
2242:
2237:
2214:
2212:
2211:
2206:
2194:
2192:
2191:
2186:
2178:
2166:
2164:
2163:
2158:
2156:
2155:
2133:
2131:
2130:
2125:
2123:
2122:
2107:
2068:
2066:
2065:
2060:
2048:
2046:
2045:
2040:
2032:
2017:
2015:
2014:
2009:
2007:
2006:
1991:
1986:
1985:
1984:
1983:
1969:
1968:
1967:
1966:
1949:
1948:
1947:
1946:
1898:
1896:
1895:
1890:
1885:
1884:
1866:
1865:
1853:
1852:
1824:
1822:
1821:
1816:
1814:
1813:
1798:
1778:
1770:
1749:
1747:
1746:
1741:
1733:
1721:
1719:
1718:
1713:
1705:
1693:
1691:
1690:
1685:
1673:
1671:
1670:
1665:
1651:
1649:
1648:
1643:
1614:
1612:
1611:
1606:
1584:
1583:
1559:
1558:
1553:
1528:
1526:
1525:
1520:
1512:
1507:
1506:
1491:
1490:
1481:
1466:
1464:
1463:
1458:
1453:
1452:
1443:
1431:
1429:
1428:
1423:
1415:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1357:
1355:
1354:
1349:
1347:
1346:
1330:unitary matrices
1328:
1326:
1325:
1320:
1301:de Bruijn graphs
1283:
1281:
1280:
1275:
1250:
1248:
1247:
1242:
1218:
1216:
1215:
1210:
1205:
1204:
1199:
1178:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1130:
1115:
1113:
1112:
1107:
1095:
1093:
1092:
1087:
1074:finite automaton
1032:machine learning
1021:
1019:
1018:
1013:
1011:
1010:
1001:
970:
968:
967:
962:
960:
959:
950:
938:
936:
935:
930:
928:
927:
918:
861:characteristic 2
853:
851:
850:
845:
840:
839:
772:unitary matrices
769:
767:
766:
761:
756:
755:
728:
726:
725:
720:
715:
714:
690:linear operators
676:
674:
673:
668:
666:
665:
638:
636:
635:
630:
625:
624:
615:
614:
605:
604:
595:
594:
569:
567:
566:
561:
510:
508:
507:
502:
494:
493:
469:
467:
466:
461:
459:
458:
430:
428:
427:
422:
408:
403:
402:
383:
381:
380:
375:
373:
372:
356:
354:
353:
348:
330:
328:
327:
322:
243:
241:
240:
235:
208:
206:
205:
200:
185:
183:
182:
177:
175:
174:
158:
156:
155:
150:
145:
144:
126:
125:
113:
112:
21:
6436:
6435:
6431:
6430:
6429:
6427:
6426:
6425:
6421:Finite automata
6406:
6405:
6404:
6399:
6371:
6321:
6310:
6283:Superconducting
6277:
6243:
6234:Neutral atom QC
6226:Ultracold atoms
6220:
6185:implementations
6184:
6178:
6118:
6111:
6078:Quantum circuit
6046:
6040:
6034:
6024:
5984:
5978:
5945:
5938:
5894:Hidden subgroup
5846:
5835:other protocols
5791:
5768:quantum network
5763:Quantum channel
5723:
5717:
5663:No-broadcasting
5653:Gottesman–Knill
5626:
5554:
5549:
5519:
5518:
5513:
5509:
5496:
5492:
5477:
5470:
5454:
5453:
5446:
5441:
5419:
5389:
5379:
5378:
5341:
5331:
5330:
5305:
5304:
5279:
5278:
5253:
5219:
5218:
5195:formal language
5164:
5154:
5153:
5138:
5085:
5080:
5079:
5049:
5044:
5043:
5007:
4988:
4960:
4955:
4954:
4916:
4915:
4887:
4886:
4853:
4822:
4821:
4775:
4770:
4769:
4750:
4749:
4722:
4721:
4714:
4685:
4667:
4648:
4624:
4603:
4588:
4583:
4582:
4531:
4530:
4508:
4507:
4483:
4473:
4472:
4451:
4446:
4445:
4424:
4419:
4418:
4399:
4398:
4379:
4378:
4354:
4341:
4328:
4302:
4301:
4279:
4278:
4257:
4252:
4251:
4220:
4207:
4192:
4164:
4159:
4158:
4130:
4125:
4124:
4099:
4094:
4093:
4068:
4063:
4062:
4038:
4028:
4027:
4002:
3997:
3996:
3971:
3966:
3965:
3940:
3935:
3934:
3899:
3883:
3873:
3872:
3850:
3849:
3822:
3821:
3793:
3776:
3763:
3758:
3757:
3733:
3728:
3727:
3706:
3701:
3700:
3679:
3674:
3673:
3638:
3584:
3579:
3578:
3548:
3543:
3542:
3515:
3510:
3509:
3484:
3479:
3478:
3459:
3458:
3430:
3413:
3396:
3379:
3374:
3373:
3342:
3337:
3336:
3322:
3292:
3287:
3286:
3265:
3260:
3259:
3238:
3233:
3232:
3211:
3206:
3205:
3179:
3169:
3156:
3143:
3135:
3134:
3099:
3089:
3088:
3064:
3054:
3053:
3032:
3031:
3026:
3020:
3019:
3014:
3004:
2992:
2991:
2967:
2962:
2961:
2940:
2939:
2934:
2928:
2927:
2922:
2912:
2898:
2893:
2892:
2868:
2867:
2862:
2856:
2855:
2850:
2840:
2826:
2821:
2820:
2787:
2759:
2734:
2733:
2723:
2720:
2719:
2709:
2702:
2694:
2693:
2654:
2648:
2647:
2623:
2610:
2605:
2604:
2602:complex numbers
2580:
2579:
2569:
2566:
2565:
2555:
2548:
2531:
2516:
2500:
2485:
2466:
2465:
2447:
2434:
2428:
2422:
2414:
2408:
2402:
2383:
2358:
2316:
2315:
2296:
2295:
2268:
2267:
2248:
2247:
2228:
2227:
2197:
2196:
2169:
2168:
2147:
2142:
2141:
2114:
2074:
2073:
2051:
2050:
2023:
2022:
1998:
1975:
1970:
1958:
1953:
1938:
1933:
1904:
1903:
1876:
1857:
1844:
1830:
1829:
1805:
1755:
1754:
1724:
1723:
1696:
1695:
1676:
1675:
1656:
1655:
1628:
1627:
1575:
1548:
1534:
1533:
1498:
1482:
1472:
1471:
1444:
1434:
1433:
1406:
1405:
1386:
1385:
1360:
1359:
1338:
1333:
1332:
1305:
1304:
1260:
1259:
1221:
1220:
1194:
1169:
1168:
1149:
1148:
1121:
1120:
1098:
1097:
1078:
1077:
1059:
1002:
992:
991:
951:
941:
940:
919:
909:
908:
831:
823:
822:
813:; it’s just an
747:
739:
738:
706:
698:
697:
657:
652:
651:
649:
616:
606:
596:
586:
572:
571:
543:
542:
536:
517:
485:
480:
479:
450:
445:
444:
394:
386:
385:
364:
359:
358:
333:
332:
313:
312:
285:graph-theoretic
281:
214:
213:
191:
190:
166:
161:
160:
136:
117:
104:
90:
89:
28:
23:
22:
15:
12:
11:
5:
6434:
6432:
6424:
6423:
6418:
6408:
6407:
6401:
6400:
6398:
6397:
6387:
6376:
6373:
6372:
6370:
6369:
6367:many others...
6364:
6359:
6354:
6349:
6340:
6326:
6324:
6316:
6315:
6312:
6311:
6309:
6308:
6303:
6298:
6293:
6287:
6285:
6279:
6278:
6276:
6275:
6270:
6265:
6260:
6254:
6252:
6245:
6244:
6242:
6241:
6239:Trapped-ion QC
6236:
6230:
6228:
6222:
6221:
6219:
6218:
6213:
6208:
6203:
6197:
6195:
6193:Quantum optics
6186:
6180:
6179:
6177:
6176:
6171:
6170:
6169:
6162:
6157:
6152:
6147:
6142:
6137:
6132:
6123:
6121:
6113:
6112:
6110:
6109:
6104:
6099:
6098:
6097:
6087:
6086:
6085:
6075:
6074:
6073:
6063:
6058:
6052:
6050:
6042:
6041:
6039:
6038:
6037:
6036:
6032:
6026:
6022:
6011:
6010:
6009:
5999:
5997:Quantum volume
5994:
5988:
5986:
5980:
5979:
5977:
5976:
5971:
5966:
5961:
5956:
5950:
5948:
5940:
5939:
5937:
5936:
5931:
5926:
5921:
5916:
5911:
5906:
5901:
5896:
5891:
5886:
5881:
5876:
5874:Boson sampling
5871:
5866:
5860:
5858:
5852:
5851:
5848:
5847:
5845:
5844:
5839:
5838:
5837:
5832:
5827:
5817:
5812:
5807:
5801:
5799:
5790:
5789:
5784:
5783:
5782:
5772:
5771:
5770:
5760:
5755:
5750:
5745:
5744:
5743:
5738:
5727:
5725:
5719:
5718:
5716:
5715:
5710:
5708:Solovay–Kitaev
5705:
5700:
5695:
5690:
5685:
5680:
5675:
5670:
5665:
5660:
5655:
5650:
5645:
5640:
5634:
5632:
5628:
5627:
5625:
5624:
5623:
5622:
5612:
5611:
5610:
5600:
5595:
5590:
5585:
5584:
5583:
5573:
5568:
5562:
5560:
5556:
5555:
5550:
5548:
5547:
5540:
5533:
5525:
5517:
5516:
5507:
5490:
5468:
5443:
5442:
5440:
5437:
5436:
5435:
5430:
5425:
5418:
5415:
5396:
5392:
5387:
5348:
5344:
5339:
5318:
5315:
5312:
5303:and the point
5292:
5289:
5286:
5260:
5256:
5252:
5249:
5246:
5243:
5240:
5237:
5234:
5230:
5227:
5190:open functions
5171:
5167:
5162:
5148:-dimensional)
5137:
5134:
5115:mixing process
5092:
5088:
5067:
5064:
5061:
5056:
5052:
5040:
5039:
5028:
5025:
5020:
5017:
5014:
5010:
5006:
5003:
5000:
4995:
4991:
4987:
4984:
4979:
4976:
4973:
4970:
4967:
4963:
4931:
4928:
4924:
4903:
4900:
4897:
4894:
4883:
4882:
4871:
4868:
4865:
4860:
4856:
4851:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4782:
4778:
4757:
4748:, measurement
4737:
4734:
4730:
4713:
4710:
4709:
4708:
4697:
4692:
4688:
4683:
4679:
4674:
4670:
4666:
4663:
4660:
4655:
4651:
4647:
4644:
4639:
4636:
4631:
4627:
4622:
4618:
4615:
4610:
4606:
4601:
4595:
4591:
4576:
4575:
4563:
4559:
4556:
4553:
4550:
4547:
4544:
4541:
4538:
4515:
4495:
4490:
4486:
4481:
4454:
4427:
4406:
4386:
4366:
4357:
4353:
4344:
4340:
4335:
4331:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4286:
4260:
4244:
4243:
4232:
4227:
4223:
4219:
4214:
4210:
4205:
4195:
4191:
4188:
4185:
4182:
4179:
4176:
4167:
4135:
4104:
4073:
4050:
4045:
4041:
4036:
4007:
3976:
3945:
3931:
3930:
3919:
3916:
3912:
3906:
3902:
3898:
3895:
3890:
3886:
3881:
3857:
3837:
3834:
3830:
3818:
3817:
3798:
3792:
3787:
3781:
3775:
3766:
3736:
3709:
3682:
3662:
3661:
3650:
3641:
3637:
3634:
3631:
3627:
3623:
3620:
3617:
3613:
3609:
3606:
3603:
3600:
3589:
3563:
3560:
3551:
3530:
3527:
3518:
3495:
3489:
3466:
3455:
3454:
3435:
3429:
3418:
3412:
3401:
3395:
3390:
3384:
3353:
3347:
3321:
3318:
3299:
3295:
3272:
3268:
3245:
3241:
3218:
3214:
3202:
3201:
3186:
3182:
3176:
3172:
3168:
3163:
3159:
3155:
3150:
3146:
3142:
3111:
3106:
3102:
3097:
3076:
3071:
3067:
3062:
3050:
3049:
3036:
3030:
3027:
3025:
3022:
3021:
3018:
3015:
3013:
3010:
3009:
3007:
3002:
2999:
2974:
2970:
2958:
2957:
2944:
2938:
2935:
2933:
2930:
2929:
2926:
2923:
2921:
2918:
2917:
2915:
2910:
2905:
2901:
2886:
2885:
2872:
2866:
2863:
2861:
2858:
2857:
2854:
2851:
2849:
2846:
2845:
2843:
2838:
2833:
2829:
2814:
2813:
2802:
2799:
2794:
2790:
2784:
2779:
2775:
2771:
2766:
2762:
2756:
2751:
2747:
2743:
2738:
2730:
2726:
2722:
2721:
2716:
2712:
2708:
2707:
2705:
2698:
2690:
2685:
2681:
2673:
2668:
2664:
2660:
2659:
2657:
2630:
2626:
2622:
2617:
2613:
2598:
2597:
2584:
2576:
2572:
2568:
2567:
2562:
2558:
2554:
2553:
2551:
2546:
2543:
2538:
2534:
2529:
2523:
2519:
2515:
2512:
2507:
2503:
2498:
2492:
2488:
2484:
2481:
2478:
2474:
2454:
2453:
2442:
2439:
2436:
2435:
2432:
2429:
2426:
2423:
2420:
2416:
2415:
2412:
2409:
2406:
2403:
2400:
2396:
2395:
2390:
2385:
2357:
2354:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2303:
2283:
2280:
2276:
2255:
2235:
2204:
2184:
2181:
2177:
2154:
2150:
2135:
2134:
2121:
2117:
2113:
2110:
2106:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2058:
2038:
2035:
2031:
2019:
2018:
2005:
2001:
1997:
1994:
1990:
1982:
1978:
1973:
1965:
1961:
1956:
1952:
1945:
1941:
1936:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1888:
1883:
1879:
1875:
1872:
1869:
1864:
1860:
1856:
1851:
1847:
1843:
1840:
1837:
1826:
1825:
1812:
1808:
1804:
1801:
1797:
1793:
1790:
1787:
1784:
1781:
1777:
1773:
1769:
1765:
1762:
1739:
1736:
1732:
1711:
1708:
1704:
1683:
1663:
1641:
1638:
1635:
1604:
1601:
1598:
1595:
1592:
1588:
1582:
1578:
1574:
1571:
1568:
1565:
1562:
1557:
1552:
1547:
1544:
1541:
1530:
1529:
1518:
1515:
1511:
1505:
1501:
1497:
1494:
1489:
1485:
1480:
1456:
1451:
1447:
1442:
1421:
1418:
1414:
1393:
1373:
1370:
1367:
1345:
1341:
1318:
1315:
1312:
1273:
1270:
1267:
1255:, carrying an
1240:
1237:
1234:
1231:
1228:
1208:
1203:
1198:
1193:
1190:
1187:
1184:
1181:
1177:
1156:
1136:
1133:
1129:
1105:
1085:
1058:
1055:
1009:
1005:
1000:
958:
954:
949:
926:
922:
917:
843:
838:
834:
830:
759:
754:
750:
746:
718:
713:
709:
705:
683:linear algebra
664:
660:
647:
628:
623:
619:
613:
609:
603:
599:
593:
589:
585:
582:
579:
559:
556:
553:
550:
534:
515:
500:
497:
492:
488:
457:
453:
420:
417:
414:
411:
407:
401:
397:
393:
371:
367:
346:
343:
340:
320:
293:directed graph
280:
277:
233:
230:
227:
224:
221:
198:
186:from a finite
173:
169:
148:
143:
139:
135:
132:
129:
124:
120:
116:
111:
107:
103:
100:
97:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6433:
6422:
6419:
6417:
6414:
6413:
6411:
6396:
6388:
6386:
6378:
6377:
6374:
6368:
6365:
6363:
6360:
6358:
6355:
6353:
6350:
6348:
6344:
6341:
6339:
6335:
6331:
6328:
6327:
6325:
6323:
6317:
6307:
6304:
6302:
6299:
6297:
6294:
6292:
6289:
6288:
6286:
6284:
6280:
6274:
6271:
6269:
6266:
6264:
6263:Spin qubit QC
6261:
6259:
6256:
6255:
6253:
6250:
6246:
6240:
6237:
6235:
6232:
6231:
6229:
6227:
6223:
6217:
6214:
6212:
6209:
6207:
6204:
6202:
6199:
6198:
6196:
6194:
6190:
6187:
6181:
6175:
6172:
6168:
6167:
6163:
6161:
6158:
6156:
6153:
6151:
6148:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6127:
6125:
6124:
6122:
6120:
6114:
6108:
6105:
6103:
6100:
6096:
6093:
6092:
6091:
6088:
6084:
6081:
6080:
6079:
6076:
6072:
6071:cluster state
6069:
6068:
6067:
6064:
6062:
6059:
6057:
6054:
6053:
6051:
6049:
6043:
6035:
6031:
6027:
6025:
6021:
6017:
6016:
6015:
6012:
6008:
6005:
6004:
6003:
6000:
5998:
5995:
5993:
5990:
5989:
5987:
5981:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5952:
5951:
5949:
5947:
5941:
5935:
5932:
5930:
5927:
5925:
5922:
5920:
5917:
5915:
5912:
5910:
5907:
5905:
5902:
5900:
5897:
5895:
5892:
5890:
5887:
5885:
5882:
5880:
5879:Deutsch–Jozsa
5877:
5875:
5872:
5870:
5867:
5865:
5862:
5861:
5859:
5857:
5853:
5843:
5840:
5836:
5833:
5831:
5828:
5826:
5823:
5822:
5821:
5818:
5816:
5815:Quantum money
5813:
5811:
5808:
5806:
5803:
5802:
5800:
5798:
5794:
5788:
5785:
5781:
5778:
5777:
5776:
5773:
5769:
5766:
5765:
5764:
5761:
5759:
5756:
5754:
5751:
5749:
5746:
5742:
5739:
5737:
5734:
5733:
5732:
5729:
5728:
5726:
5724:communication
5720:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5694:
5691:
5689:
5686:
5684:
5681:
5679:
5676:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5636:
5635:
5633:
5629:
5621:
5618:
5617:
5616:
5613:
5609:
5606:
5605:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5582:
5579:
5578:
5577:
5574:
5572:
5569:
5567:
5564:
5563:
5561:
5557:
5553:
5546:
5541:
5539:
5534:
5532:
5527:
5526:
5523:
5511:
5508:
5504:
5500:
5494:
5491:
5487:
5483:
5482:
5475:
5473:
5469:
5464:
5463:
5458:
5455:Kondacs, A.;
5451:
5449:
5445:
5438:
5434:
5433:Real computer
5431:
5429:
5426:
5424:
5421:
5420:
5416:
5414:
5412:
5394:
5390:
5376:
5372:
5368:
5365:given by the
5364:
5346:
5342:
5313:
5287:
5276:
5258:
5247:
5241:
5232:
5216:
5210:
5208:
5204:
5200:
5196:
5191:
5187:
5169:
5165:
5151:
5147:
5143:
5135:
5133:
5131:
5127:
5126:stack machine
5123:
5118:
5116:
5112:
5106:
5090:
5086:
5062:
5054:
5050:
5026:
5023:
5018:
5015:
5012:
5008:
5001:
4993:
4989:
4985:
4982:
4974:
4968:
4965:
4961:
4953:
4952:
4951:
4949:
4945:
4926:
4898:
4892:
4869:
4866:
4858:
4854:
4842:
4836:
4833:
4830:
4827:
4820:
4819:
4818:
4817:
4813:
4808:
4806:
4802:
4798:
4780:
4776:
4755:
4732:
4719:
4711:
4690:
4686:
4672:
4668:
4664:
4661:
4658:
4653:
4649:
4642:
4637:
4634:
4629:
4625:
4620:
4616:
4608:
4604:
4593:
4589:
4581:
4580:
4579:
4554:
4551:
4545:
4542:
4539:
4536:
4529:
4528:
4527:
4513:
4488:
4484:
4452:
4425:
4384:
4355:
4351:
4342:
4338:
4333:
4329:
4325:
4322:
4319:
4313:
4310:
4298:
4284:
4258:
4247:
4230:
4225:
4208:
4193:
4186:
4180:
4174:
4165:
4157:
4156:
4155:
4039:
3914:
3904:
3900:
3896:
3884:
3871:
3870:
3869:
3855:
3832:
3785:
3773:
3764:
3756:
3755:
3754:
3734:
3707:
3680:
3671:
3667:
3639:
3635:
3629:
3621:
3615:
3604:
3601:
3598:
3577:
3576:
3575:
3561:
3558:
3549:
3528:
3525:
3516:
3493:
3464:
3427:
3410:
3393:
3388:
3372:
3371:
3370:
3369:
3351:
3335:
3334:Hilbert space
3330:
3328:
3319:
3317:
3315:
3314:de Rham curve
3297:
3293:
3270:
3266:
3243:
3239:
3216:
3212:
3184:
3174:
3166:
3161:
3157:
3148:
3144:
3133:
3132:
3131:
3129:
3125:
3104:
3100:
3069:
3065:
3034:
3028:
3023:
3016:
3011:
3005:
3000:
2997:
2990:
2989:
2988:
2972:
2968:
2942:
2936:
2931:
2924:
2919:
2913:
2908:
2903:
2899:
2891:
2890:
2889:
2870:
2864:
2859:
2852:
2847:
2841:
2836:
2831:
2827:
2819:
2818:
2817:
2800:
2797:
2792:
2788:
2782:
2777:
2773:
2769:
2764:
2760:
2754:
2749:
2745:
2741:
2736:
2728:
2724:
2714:
2710:
2703:
2696:
2688:
2683:
2679:
2671:
2666:
2662:
2655:
2646:
2645:
2644:
2628:
2624:
2620:
2615:
2611:
2603:
2582:
2574:
2570:
2560:
2556:
2549:
2544:
2536:
2532:
2521:
2517:
2513:
2505:
2501:
2490:
2486:
2482:
2476:
2464:
2463:
2462:
2461:
2451:
2446:
2445:State Diagram
2440:
2417:
2397:
2394:
2389:
2381:
2380:
2377:
2372:
2371:
2368:
2367:
2364:given by the
2363:
2355:
2353:
2336:
2330:
2327:
2324:
2321:
2301:
2278:
2253:
2225:
2220:
2218:
2202:
2179:
2152:
2148:
2138:
2119:
2108:
2100:
2094:
2082:
2079:
2072:
2071:
2070:
2033:
2003:
1992:
1980:
1976:
1971:
1963:
1959:
1954:
1950:
1943:
1939:
1934:
1930:
1924:
1918:
1912:
1909:
1902:
1901:
1900:
1881:
1877:
1873:
1870:
1867:
1862:
1858:
1854:
1849:
1845:
1838:
1835:
1810:
1799:
1791:
1785:
1779:
1771:
1763:
1753:
1752:
1751:
1734:
1706:
1681:
1661:
1654:
1639:
1636:
1633:
1625:
1620:
1618:
1593:
1590:
1580:
1576:
1569:
1563:
1555:
1542:
1513:
1503:
1499:
1495:
1483:
1470:
1469:
1468:
1445:
1416:
1391:
1368:
1365:
1343:
1339:
1331:
1316:
1313:
1310:
1302:
1298:
1294:
1289:
1287:
1268:
1258:
1257:inner product
1254:
1251:-dimensional
1235:
1232:
1229:
1201:
1188:
1185:
1179:
1167:-state qudit
1154:
1131:
1119:
1103:
1083:
1075:
1070:
1068:
1064:
1056:
1054:
1052:
1047:
1045:
1041:
1037:
1033:
1029:
1023:
1007:
1003:
989:
986:
982:
978:
974:
956:
952:
924:
920:
907:
902:
900:
896:
892:
888:
884:
879:
877:
873:
869:
864:
862:
858:
836:
832:
820:
816:
812:
808:
804:
800:
795:
793:
789:
788:automorphisms
785:
781:
777:
773:
752:
748:
736:
732:
711:
707:
695:
691:
686:
684:
680:
662:
658:
646:
642:
626:
621:
617:
611:
607:
601:
597:
591:
587:
583:
580:
577:
557:
554:
551:
548:
540:
533:
529:
525:
521:
514:
498:
495:
490:
486:
477:
473:
455:
451:
442:
438:
434:
412:
409:
399:
395:
369:
365:
341:
338:
308:
306:
302:
301:column vector
298:
294:
290:
286:
278:
276:
274:
270:
266:
262:
258:
254:
249:
247:
228:
222:
219:
212:
189:
171:
167:
141:
137:
133:
130:
127:
122:
118:
114:
109:
105:
98:
95:
88:
83:
81:
77:
73:
72:Markov chains
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
6291:Charge qubit
6216:KLM protocol
6165:
6029:
6019:
5713:Purification
5643:Eastin–Knill
5510:
5502:
5497:I. Baianu, "
5493:
5485:
5479:
5460:
5411:metric space
5374:
5370:
5214:
5211:
5198:
5145:
5139:
5119:
5107:
5041:
4884:
4809:
4715:
4577:
4299:
4248:
4245:
3932:
3819:
3669:
3663:
3574:, such that
3456:
3331:
3323:
3203:
3051:
2959:
2887:
2815:
2599:
2457:
2444:
2392:
2387:
2375:
2359:
2221:
2139:
2136:
2020:
1899:is given by
1827:
1624:accept state
1621:
1531:
1290:
1284:that is the
1071:
1060:
1048:
1024:
981:mixed states
903:
899:Markov chain
880:
865:
818:
810:
802:
796:
780:distribution
778:can be some
775:
730:
693:
687:
678:
644:
538:
531:
523:
519:
512:
475:
471:
440:
436:
432:
309:
282:
272:
250:
246:accept state
84:
79:
75:
64:measure-many
63:
60:measure-once
59:
43:
39:
35:
29:
6322:programming
6301:Phase qubit
6206:Circuit QED
5678:No-deleting
5620:cloud-based
5505:pp.339-354.
5457:Watrous, J.
5273:. But this
5203:M-automaton
4816:mixed state
4795:, such the
4140:non-halting
4012:non-halting
3670:non-halting
3666:linear span
3440:non-halting
735:unit vector
526:th row. By
211:probability
159:of letters
6410:Categories
6362:libquantum
6296:Flux qubit
6201:Cavity QED
6150:Bacon–Shor
6140:stabilizer
5668:No-cloning
5217:; that is
5186:isometries
4812:pure state
4803:and other
1063:Cris Moore
1051:Ion Baianu
868:vice versa
737:, and the
439:states in
6268:NV center
5703:Threshold
5683:No-hiding
5648:Gleason's
5317:⟩
5314:ψ
5291:⟩
5251:⟩
5248:ψ
5239:⟨
5091:α
5055:α
5013:α
4994:α
4986:∫
4975:ρ
4966:α
4930:⟩
4927:ψ
4864:⟩
4855:ψ
4834:∫
4828:ρ
4781:α
4736:⟩
4733:ψ
4696:⟩
4662:α
4643:δ
4635:∈
4621:∑
4614:⟩
4594:α
4558:→
4552:×
4549:Σ
4546:×
4537:δ
4514:δ
4494:⟩
4405:Σ
4323:δ
4317:Σ
4285:κ
4222:‖
4218:⟩
4213:′
4209:ψ
4190:‖
4181:σ
4175:
4049:⟩
4044:′
4040:ψ
3918:⟩
3915:ψ
3905:α
3894:⟩
3889:′
3885:ψ
3856:α
3836:⟩
3833:ψ
3791:→
3636:∈
3633:⟩
3619:⟩
3605:
3559:⊆
3526:⊆
3428:⊕
3411:⊕
3185:∗
3175:∗
3162:∗
3149:∗
3110:⟩
3075:⟩
2783:∗
2755:∗
2689:∗
2672:∗
2600:with the
2542:⟩
2511:⟩
2480:⟩
2477:ψ
2337:σ
2331:
2325:≤
2302:σ
2282:⟩
2279:ψ
2234:Σ
2203:σ
2183:⟩
2180:ψ
2153:α
2116:‖
2112:⟩
2109:ψ
2098:‖
2089:∅
2083:
2057:∅
2037:⟩
2034:ψ
2000:‖
1996:⟩
1993:ψ
1977:σ
1960:σ
1951:⋯
1940:σ
1928:‖
1919:σ
1913:
1878:σ
1871:⋯
1859:σ
1846:σ
1836:σ
1807:‖
1803:⟩
1800:ψ
1789:‖
1783:⟩
1780:ψ
1764:ψ
1761:⟨
1738:⟩
1735:ψ
1710:⟩
1707:ψ
1637:×
1597:Σ
1594:∈
1591:α
1581:α
1567:Σ
1517:⟩
1514:ψ
1504:α
1493:⟩
1488:′
1484:ψ
1455:⟩
1450:′
1446:ψ
1420:⟩
1417:ψ
1392:α
1372:Σ
1369:∈
1366:α
1344:α
1314:×
1272:‖
1269:⋅
1266:‖
1233:−
1186:∈
1183:⟩
1180:ψ
1135:⟩
1132:ψ
872:languages
837:α
807:power set
753:α
712:α
663:α
612:α
602:β
592:γ
584:⋯
558:⋯
555:γ
552:β
549:α
496:∈
456:α
416:Σ
413:∈
410:α
400:α
370:α
345:Σ
342:∈
339:α
319:Σ
253:languages
229:σ
223:
197:Σ
168:σ
138:σ
131:⋯
119:σ
106:σ
96:σ
6330:OpenQASM
6306:Transmon
6183:Physical
5983:Quantum
5884:Grover's
5658:Holevo's
5631:Theorems
5581:timeline
5571:NISQ era
5417:See also
5377:, where
4578:so that
4377:. Here,
2224:language
1042:and the
784:manifold
357:, write
188:alphabet
6320:Quantum
6258:Kane QC
6117:Quantum
6045:Quantum
5974:PostBQP
5944:Quantum
5929:Simon's
5722:Quantum
5559:General
5111:ergodic
3664:is the
2960:Taking
2441:
2356:Example
1615:form a
1116:-state
895:simplex
6338:IBM QX
6334:Qiskit
6273:NMR QC
6251:-based
6155:Steane
6126:Codes
5924:Shor's
5830:SARG04
5638:Bell's
5363:metric
4799:, the
4109:reject
4078:accept
3981:reject
3950:accept
3803:accept
3594:accept
3423:reject
3406:accept
2384:State
530:, let
87:string
6160:Toric
5603:Qubit
5439:Notes
5373:or a
2888:and
1118:qudit
782:on a
733:by a
50:or a
42:) or
6352:Cirq
6343:Quil
6249:Spin
6145:Shor
5825:BB84
5758:LOCC
4444:and
3726:and
3602:span
3541:and
3332:The
3285:and
3231:and
1622:The
1291:The
1065:and
857:ring
679:etc.
537:and
251:The
62:and
6166:gnu
6130:CSS
6007:XEB
5969:QMA
5964:QIP
5959:EQP
5954:BQP
5934:VQE
5889:HHL
5693:PBR
5486:237
5329:in
5124:or
4457:rej
4430:acc
4360:rej
4347:acc
4263:non
4198:acc
4170:acc
4123:or
4092:or
3769:acc
3739:non
3712:rej
3699:,
3685:acc
3644:acc
3554:rej
3521:acc
3087:or
2167:on
1299:or
990:on
859:of
817:on
809:of
770:by
650:by
474:by
470:is
267:of
259:of
78:or
40:QFA
30:In
6412::
6357:Q#
5503:33
5484:,
5471:^
5447:^
5209:.
5105:.
4526:,
4417:,
4397:,
4166:Pr
3995:,
3964:,
3158:01
3130::
2352:.
2328:Pr
2222:A
2080:Pr
1910:Pr
1619:.
1467::
1295:,
1288:.
1053:.
1022:.
901:.
863:.
685:.
677:,
524:q'
307:.
220:Pr
82:.
34:,
6345:–
6336:–
6332:–
6033:2
6030:T
6023:1
6020:T
5544:e
5537:t
5530:v
5395:N
5391:P
5386:C
5347:N
5343:P
5338:C
5311:|
5288:P
5285:|
5259:2
5255:|
5245:|
5242:P
5236:|
5233:=
5229:r
5226:P
5215:P
5170:N
5166:P
5161:C
5146:N
5087:U
5066:)
5063:x
5060:(
5051:p
5027:x
5024:d
5019:x
5016:,
5009:U
5005:)
5002:x
4999:(
4990:p
4983:=
4978:)
4972:(
4969:,
4962:U
4923:|
4902:)
4899:x
4896:(
4893:p
4870:x
4867:d
4859:x
4850:|
4846:)
4843:x
4840:(
4837:p
4831:=
4777:U
4756:P
4729:|
4691:j
4687:q
4682:|
4678:)
4673:j
4669:q
4665:,
4659:,
4654:i
4650:q
4646:(
4638:Q
4630:j
4626:q
4617:=
4609:i
4605:q
4600:|
4590:U
4562:C
4555:Q
4543:Q
4540::
4489:0
4485:q
4480:|
4453:Q
4426:Q
4385:Q
4365:)
4356:Q
4352:;
4343:Q
4339:;
4334:0
4330:q
4326:;
4320:;
4314:;
4311:Q
4308:(
4259:P
4231:,
4226:2
4204:|
4194:P
4187:=
4184:)
4178:(
4134:H
4103:H
4072:H
4035:|
4006:H
3975:H
3944:H
3911:|
3901:U
3897:=
3880:|
3829:|
3797:H
3786:Q
3780:H
3774::
3765:P
3735:P
3708:P
3681:P
3649:}
3640:Q
3630:q
3626:|
3622::
3616:q
3612:|
3608:{
3599:=
3588:H
3562:Q
3550:Q
3529:Q
3517:Q
3494:Q
3488:H
3465:Q
3434:H
3417:H
3400:H
3394:=
3389:Q
3383:H
3352:Q
3346:H
3298:1
3294:U
3271:0
3267:U
3244:2
3240:a
3217:1
3213:a
3181:)
3171:)
3167:0
3154:(
3145:1
3141:(
3105:2
3101:S
3096:|
3070:1
3066:S
3061:|
3035:]
3029:0
3024:0
3017:0
3012:1
3006:[
3001:=
2998:P
2973:1
2969:S
2943:]
2937:1
2932:0
2925:0
2920:1
2914:[
2909:=
2904:1
2900:U
2871:]
2865:0
2860:1
2853:1
2848:0
2842:[
2837:=
2832:0
2828:U
2801:1
2798:=
2793:2
2789:a
2778:2
2774:a
2770:+
2765:1
2761:a
2750:1
2746:a
2742:=
2737:]
2729:2
2725:a
2715:1
2711:a
2704:[
2697:]
2684:2
2680:a
2667:1
2663:a
2656:[
2629:2
2625:a
2621:,
2616:1
2612:a
2583:]
2575:2
2571:a
2561:1
2557:a
2550:[
2545:=
2537:2
2533:S
2528:|
2522:2
2518:a
2514:+
2506:1
2502:S
2497:|
2491:1
2487:a
2483:=
2473:|
2433:1
2431:S
2427:2
2425:S
2421:2
2419:S
2413:2
2411:S
2407:1
2405:S
2401:1
2399:S
2393:0
2388:1
2340:)
2334:(
2322:p
2275:|
2254:p
2176:|
2149:U
2120:2
2105:|
2101:P
2095:=
2092:)
2086:(
2030:|
2004:2
1989:|
1981:0
1972:U
1964:1
1955:U
1944:k
1935:U
1931:P
1925:=
1922:)
1916:(
1887:)
1882:k
1874:,
1868:,
1863:1
1855:,
1850:0
1842:(
1839:=
1811:2
1796:|
1792:P
1786:=
1776:|
1772:P
1768:|
1731:|
1703:|
1682:N
1662:P
1640:N
1634:N
1603:)
1600:}
1587:|
1577:U
1573:{
1570:,
1564:,
1561:)
1556:N
1551:C
1546:(
1543:P
1540:(
1510:|
1500:U
1496:=
1479:|
1441:|
1413:|
1340:U
1317:N
1311:N
1239:)
1236:1
1230:N
1227:(
1207:)
1202:N
1197:C
1192:(
1189:P
1176:|
1155:N
1128:|
1104:N
1084:N
1008:N
1004:P
999:C
957:N
953:P
948:C
925:N
921:P
916:C
842:}
833:U
829:{
819:Q
811:Q
803:q
776:q
758:}
749:U
745:{
731:q
717:}
708:U
704:{
694:q
659:U
648:0
645:q
627:.
622:0
618:q
608:U
598:U
588:U
581:=
578:q
539:q
535:0
532:q
520:q
516:0
513:q
499:Q
491:0
487:q
476:N
472:N
452:U
441:Q
437:N
433:Q
419:}
406:|
396:U
392:{
366:U
232:)
226:(
172:i
147:)
142:k
134:,
128:,
123:1
115:,
110:0
102:(
99:=
38:(
20:)
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