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[Online].Available:https://doi.org/10.1038/s41467-021-22539-9 |
distributedquantumcomputingenvironments.Finally,wehave [15] “IBMQuantum,”https://www.ibm.com/quantum,accessed:2022-07-30. |
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Quantum Chaos & Quantum Computers |
∗ |
D. L. Shepelyansky |
Laboratoire de Physique Quantique, UMR 5626 du CNRS, Universit´e Paul Sabatier, F-31062 Toulouse Cedex 4, France |
(June 16, 2000) |
The standard generic quantum computer model is studied analytically and numerically and the |
borderforemergenceofquantumchaos,inducedbyimperfectionsandresidualinter-qubitcouplings, |
is determined. This phenomenon appears in an isolated quantum computer without any external |
decoherence. The onset of quantum chaos leads to quantum computer hardware melting, strong |
quantumentropygrowth anddestruction ofcomputeroperability. Thetimescales for development |
of quantum chaos and ergodicity are determined. In spite the fact that this phenomenon is rather |
dangerousforquantumcomputingitisshownthatthequantumchaosborderforinter-qubitcoupling |
is exponentially larger than the energy level spacing between quantum computer eigenstates and |
0002 |
drops only linearly with the number of qubits n. As a result the ideal multi-qubit structure of the |
computer remains rather robust against imperfections. This opens a broad parameter region for a |
possible realization of quantum computer. The obtained results are related to the recent studies of |
quantum chaos in such many-body systems as nuclei, complex atoms and molecules, finite Fermi |
nuJ |
systems and quantumspin glass shards which are also reviewed in thepaper. |
PACS numbers: 03.67.Lx, 05.45.Mt, 24.10.Cn |
51 |
Lecture at Nobel symposium on “Quantum chaos”, June 2000, Sweden |
1v3706000/hp-tnauq:viXra |
I. INTRODUCTION half spins (see recent review [2] and references there in). |
The computer operation is based on controlled series of |
On the border between two Millennia it is natural to two-qubitcouplingswitchonandoffwhichtogetherwith |
ask a question, what will be the origin of future human one-qubitrotationsallowtorealizeanyunitaryoperation |
power? Eventhirtyortwentyyearsagothestandardan- intheHilbertspaceofsizeN H =2n [3,4]. Inthisrespect |
swer would be: nuclear. But now in a view of amazing theinter-qubitcouplingbecomesunavoidablepropertyof |
computerdevelopmentallovertheworlditbecomesclear quantum computer. |
thatthefuturepowerwillberelatedtoabilitytocountas Recently a great increase of interest to quantum com- |
fastaspossible. InIMillenniumthisabilitywasbasically puting has been generated by the work of Shor [5] who |
comparable with finger counting, while at the end of II constructed a quantum algorithm which performs large |
Millennium it made enormous jump with computer cre- number factorization into primes exponentially faster |
ationwhichledtoaqualitativechangeinhumansociety. thananyknownclassicalalgorithm. AlsoGrovershowed |
During lasttwo decades the power ofmodern computers [6] that a search of an item in a long list is done much |
demonstrated a constant impressive growth due to tech- faster by quantum computer. The enormous gain in |
nological progress and creation of chips of smaller and the computation rate is reached due to high parallelism |
smaller size. In a near future this size should reach a of multi-qubit quantum evolution and quantum interfer- |
scale at which the quantum nature of physical laws will ence. Together with a recent theoretical development of |
become dominant. As a result, we unavoidably come to quantum error-correcting codes [7,8] these exciting re- |
thecreationproblemofquantumcomputer. Suchacom- sults stimulated various experimental proposals for re- |
puter shouldbe essentiallybasedonquantummechanics alization of quantum computer. The variety of phys- |
and operate with unitary transformations and quantum ical systems proposed is really amazing and includes: |
logic. The unitary nature of transformations allows to ion traps [9], nuclear magnetic resonance systems [10], |
exclude energy dissipation that should play an impor- nuclear spins with interaction controlled electronically |
tant role on small scales. At the same time, as stressed [11,12]orbylaserpulses[13],quantumdots[14],Cooper |
byFeynman[1],theclassicalcomputerhasenormousdif- pair boxes [15], optical lattices [16] and electrons float- |
ficultyinsimulationofmany-bodyquantumsystemsdue ing on liquid helium [17]. At present two-qubit gates |
toexponentialgrowthoftheHilbertspacewiththenum- were experimentally realized with cold atoms [18], and |
ber of particles and hence, of the computational efforts. theGroveralgorithmhasbeenperformedforthreequbits |
Duetothatitispossibletoexpectthatacomputercom- made from nuclear spins in a molecule [19]. |
posed of quantum elements will be much more efficient Thustherearetwomainlinesinthepresentdayquan- |
forsolutionofquantum,andmaybeother,problems. At tum computer research: construction and development |
presentquantum computer is viewed as a systemof cou- of efficient quantum algorithms and searchfor a optimal |
pled n qubits being two-level quantum systems or one- physicalsystemwithafuture experimentalrealizationof |
few coupled qubits. The first line has a strong mathe- |
1 |
maticalshade: indeed,itassumesthatallqubitsareper- modified resulting in destruction of quantum computer |
fectly identical and the couplings between them can be hardware. In spite of this expectation it has been shown |
operatedalsoinaperfectway. Thesecondline,inalarge recently that the ideal qubit structure is much more ro- |
respect,isapartofexperimentalphysicswithfewqubits. bust and in reality the quantum hardware melting and |
As a result, there is a broadopen field for physicalstud- quantum chaos induced by inter-qubit interaction takes |
ies of a realistic quantum computer with many qubits. place at J >J ∆0/n being exponentially larger than |
c |
∼ |
Indeed, in reality the qubits are never perfect, the level ∆ [24]. This result for quantum chaos border in quan- |
n |
spacing fluctuates from one qubit to another due to dif- tum computing is recently confirmed by more extended |
ferentenvironment,thereisalsoaresidualinteractionbe- studies [25] and opens a broad regime of parameters for |
tweenqubits whichcannotbeeliminatedcompletely,the whichrealizationofaquantumcomputerispossible. For |
two-qubit gate operations are also not perfect. In prac- example,at∆0 1Kandn=1000the criticalcoupling |
∼ |
tice,atleastaroundn=1000ofsuchqubitsarerequired J 1mK iscompatiblewiththe experimentalproposal |
c |
∼ |
to makeaquantumcomputer moreefficientthanthe ex- [12]. |
isting our days computers [2]. In addition to the above The above result is closely related to the long term |
internal imperfections there is also decoherence due to research of quantum many-body systems, started by |
couplingtoexternalworldwhichproducesnoiseanddis- Wigner [26] interested in “the properties of the wave |
sipation. The effects of decoherence and two-qubit gates functions of quantum mechanical systems which are as- |
pulsebroadeningwerenumericallytestedonShor’salgo- sumed to be so complicated that statistical considera- |
rithm [20,21]and were shown to play an important role. tion can be applied to them”. As a result, the ran- |
Atthesametimetheestimatesshowthatitispossibleto dommatrixtheory(RMT)hasbeendevelopedtoexplain |
havephysicalqubitswithverylongrelaxationtime, dur- the generic properties ofcomplex eigenstates andenergy |
ingwhichmanygateoperationscanberealized[22]. One spectra of many-body interacting systems such as heavy |
of the most promising system looks to be nuclear spins nuclei,manyelectronatomsandmolecules[27]. Thesuc- |
in two-dimensional semiconductor structures [11,12,23]. cess of RMT was so impressive [28] that the conditions |
However, the absence of external decoherence does ofitsapplicabilitytomany-bodysystemswerenotreally |
not yet mean that the computer will operate properly. realizeduntil recently. Indeed, forexample, in nucleithe |
Indeed, internal imperfections with inter-qubit residual density of states growsexponentially with excitation en- |
couplings J can strongly modify the ideal quantum reg- ergy and at a first glance the RMT is valid as soon as |
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