WO2016020648A2 - Correction d'erreur passive d'erreurs de logique quantique - Google Patents

Correction d'erreur passive d'erreurs de logique quantique Download PDF

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WO2016020648A2
WO2016020648A2 PCT/GB2015/052199 GB2015052199W WO2016020648A2 WO 2016020648 A2 WO2016020648 A2 WO 2016020648A2 GB 2015052199 W GB2015052199 W GB 2015052199W WO 2016020648 A2 WO2016020648 A2 WO 2016020648A2
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qubit
quantum
qubits
error
logic system
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WO2016020648A3 (fr
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Eliot KAPIT
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Isis Innovation Limited
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B82—NANOTECHNOLOGY
    • B82Y—SPECIFIC USES OR APPLICATIONS OF NANOSTRUCTURES; MEASUREMENT OR ANALYSIS OF NANOSTRUCTURES; MANUFACTURE OR TREATMENT OF NANOSTRUCTURES
    • B82Y10/00—Nanotechnology for information processing, storage or transmission, e.g. quantum computing or single electron logic
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena

Definitions

  • This invention relates to a quantum logic system arranged to perform passive error correction of quantum logic errors.
  • this invention relates to a quantum logic system for use in quantum computing memory systems.
  • a quantum bit also known as a qubit
  • the qubit is similar to a classical bit in that it can adopt a value of 0 or 1, but different from the classical bit in that it can adopt both values simultaneously (in a state known as superposition).
  • the computational power of a quantum computer lies in the superposition.
  • Qubits are implemented in two state quantum mechanical systems (the states corresponding to 0 and 1).
  • qubits may be implemented by trapping photons, trapping electrons, controlling electron spin, or controlling nuclear spin.
  • the difference between the states is small (e.g. the presence or a microwave photon or an electron) and therefore qubits can be susceptible to quantum errors, where interaction with a surrounding environment causes the flip of one or more qubits. Shielding can mitigate but not completely eradicate errors, and so error correction protocols are necessary for a successful quantum computation.
  • a quantum logic system arranged to perform passive error correction .
  • the quantum logic system may comprise a plurality of qubits, the qubits may be arranged into a plurality of qubit groups and may have a first decay rate .
  • the quantum logic system may further comprise control circuitry.
  • the control circuitry may passively constrain the overall state of each qubit group .
  • the quantum logic system may also comprise a plurality of auxiliary quantum obj ects.
  • the auxiliary quantum obj ects may be resonantly coupled to the qubit groups such that errors in any of the plurality of qubits are resonantly transferred to the auxiliary quantum objects.
  • the auxiliary quantum objects may have a second decay rate, which may be faster than the first decay rate such that qubit errors resonantly transferred to the auxiliary quantum objects are more likely to decay at the auxiliary quantum objects rather than at the qubit from which they are transferred. This may correct the quantum error.
  • At least some embodiments of the quantum logic system are advantageous because the errors are corrected without external interference (passive error correction). This means that the logical error rate in the quantum logic system can be significantly reduced without the need for resource and time intensive algorithmic error correcting techniques.
  • the quantum logic system can still be integrated with algorithmic error correction systems, but the requirements of the algorithmic system are significantly reduced since the passive error correction eliminates the majority of errors.
  • At least some embodiments are also advantageous because the quantum logic system is scalable to very large qubit systems.
  • the machine readable medium referred to herein may be any of the following: a floppy disk, a CD-ROM/ RAM, a DVD ROM (including -R/-RW or +R/+RW), DVD RAM, a memory (such as an SD card, a Flash drive, a CF card, or the like), a hard drive (including a platter based drive, or a Solid State Drive), a tape, an Internet Download (including a FTP transfer, an http transfer, or the like); a wire.
  • Figure 1 is a schematic representation of a qubit system with topological order
  • Figure 2 shows the system of Figure 1 , with a single qubit error
  • Figure 3 shows the central portion (A) of the system of Figure 1 , showing the shadow lattice coupled to alternate plaquettes;
  • Figure 4 shows the system of Figure 3, with multiple shadow qubits coupled to each plaquette
  • Figure 5A shows the system of Figure 2 with three adjacent qubit errors
  • Figure 5B shows the system of Figure 2 following a subsequent qubit error
  • Figure 6 shows the energy profile for the secondary repair qubits
  • Figure 7 shows a simple three qubit system to illustrate the mechanism of passive error correction
  • Figures 8A-C show simulations of logical state lifetimes for different lattice parameters
  • Figure 9 shows a simulation of logical state lifetimes for different primary repair qubit parameters
  • Figure 10 shows simulations of logical state lifetimes of a system with physical edges
  • Figure 11 shows the three body gadget discussed in Appendix B
  • Figure 12 shows the Eigensystem of the 3-body gadget of Figure 10
  • Figure 13 shows the four body gadget discussed in Appendix B
  • Figure 14 shows the Eigensystem of the 4-body gadget of Figure 12
  • Figure 15 shows the five body gadget discussed in Appendix B.
  • a many-body quantum system (a system including many qubits representing the logical states of a computer) is coupled to an array of "bad" quantum objects with fast decay rates (the "shadow lattice") .
  • Unwanted many-body excitations errors in one or more qubits
  • the resonant energy transfer ensures that errors in the qubits are eliminated rapidly, while minimising the creation of errors by the shadow lattice itself.
  • a quantum object is considered to be a "bad” quantum object when it has a decay rate that is faster than the decay rate of the qubits.
  • the decay rate of the "bad" quantum objects is considered to be fast when it is substantially at least an order of magnitude faster than the decay rates of the qubits, such that the errors decay in the "bad” quantum objects.
  • the decay rate of the quantum objects may be more or less than an order of magnitude faster than the decay rate of the qubits.
  • passive error correction is performed on a system 100 based on Kitaev's toric code disclosed in A Kitaev, Annals of Physics 303, 230 (2003).
  • Figure 1 shows a section of the system 100.
  • the system 100 displays topological order and has a primary lattice of primary qubits 102 arranged on the midpoints between vertices in a square lattice .
  • the primary qubits 102 are arranged into groups 104 of four qubits 102 in a diamond shape (the shaded regions) .
  • Each qubit 102 is shared between two groups 104 such that groups 104 only meet at their vertices and not along their edges.
  • the groups 104 are known as plaquettes, and the space between the groups (the white regions) are known as Stars 1 10. In other embodiments, different arrangements of qubits may be provided.
  • Figure 1 shows sixteen complete plaquettes 104 and nine complete stars 1 10. However, the arrangement of Figure 1 is representative of a larger system formed of repeating plaquettes 104 and stars 1 10.
  • the total system 100 is arranged as a toroid, such that there are no edge effects and the overall system 100 contains L by L plaquettes 104 and the same number of stars 1 10.
  • a toroid is formed by joining opposing edges, such that a tube is formed, with the ends of the tube being joined to form a ring.
  • a, b, c and d are the states (0 or 1) of each of the qubits 102 in the plaquette 104.
  • the qubit states (a, b, c, d) are chosen to be in the s x basis of each qubit.
  • Star operators take the same form, but act in the (orthogonal) s y basis.
  • Each plaquette 104 is provided with circuitry (not shown Figure 1) that constrains the overall state of the plaquette 104 to be in a logical state. An example of such circuitry is discussed in Appendix B .
  • the stars 1 10 are provided with similar circuitry to constrain the state of the stars 1 10.
  • Figure 2 shows the system 100 of Figure 1 , when a single random qubit error has occurred on a first qubit 102a.
  • the qubit error (of the s y type) causes the state of the qubit 102a to flip to 1 and if the qubit 102a is initially in state 1 , the qubit error causes the state of the qubit 102a to flip to 0.
  • the system Since the state of one of the qubits has changed, the system is in a first error state .
  • the first error state can still be corrected to the true logical state by algorithmic error correction, however, a total of N qubit errors would eventually lead to a logical error. Without algorithmic error correction, a single qubit error could lead to a logical error.
  • the first qubit 102a is shared between a first plaquette 104a and a second plaquette 104b .
  • the qubit error causes the state of the first 104a and second 104b plaquettes to flip, creating a pair of plaquette errors .
  • Each plaquette error is also known as an anyon.
  • the circuitry biases the plaquettes 104 such that the energy cost to flip the state of a plaquette 104 is V. Therefore, the cost of a qubit error on the first qubit 102a, which creates two plaquette errors is 2V.
  • Figure 3 shows the central portion A of the system 100 of Figure 1 .
  • alternate plaquettes 104 of the system 100 are coupled to separate shadow qubit systems 1 12, to provide passive error correction.
  • Each shadow qubit system 1 12 includes a first primary repair qubit 1 14a.
  • the first primary repair qubit 1 14a is intentionally formed with a fast decay rate, and the energy to excite the first primary repair qubits 1 14a is tuned to be in resonance with the creation of two qubit errors .
  • the energy required to excited the first primary repair qubit 1 14a may be the same as the energy to create two plaquette errors .
  • the energy to excite the primary repair qubit l l 4a may be the same as the energy to create two plaquette errors plus or minus the frequency of one of the applied fields .
  • This tuning means that when an error forms on the first qubit 102a, it is resonantly transferred to the associated first primary repair qubit 1 14ai.
  • the primary repair qubits also correct star errors in the same fashion.
  • the resonant transfer causes the first qubit 102a to flip from its error state, back to its original state and causes the associated first primary repair qubit 1 14ai to flip from its ground state to an excited state.
  • the logical state is only concerned with the primary qubits 102 and so resonant transfer causes the system 100 to revert back to the correct logical state.
  • each shadow qubit system 1 12 may be provided with a number of repair qubits 1 14 tuned to be in resonance with different error mechanisms, as shown in Figure 4.
  • FIG 4 shows the section A of the system 100 of Figure 1 , with each shadow qubit system 1 12 having 9 repair qubits 1 14.
  • each shadow qubit system 1 12 there is a first primary repair qubit 1 14a as discussed in relation to Figure 3.
  • the other shadow qubits 1 14b-i are tuned to correct the further error process, which will be discussed in relation to Figures 5A and 5B .
  • the shadow qubits 1 14 form a shadow lattice .
  • Figure 5 A shows the system 100 of Figure 1 , where errors have occurred in both the second 102b and third 102c qubits. No error has occurred in the first qubit 102a.
  • the first 102a, second 102b and third 102c qubits are provided in a single row of the lattice, adjacent to one another, with the second 120b and third 102c qubits either side of the first 102a.
  • the pair of qubit errors causes errors in the first 104a, second 104b, third 104c and fourth 104d plaquettes, which are adjacent to one another.
  • alternate plaquettes 104 are provided with shadow qubit systems 1 14. Therefore, the third 104c and second 104b plaquettes will be provided with shadow qubits systems 1 14 1 , 1 14 2 .
  • the first primary repair qubits 1 14ai, 1 14a 2 are equally likely to flip the first 102a, second 102b or third 102c qubits to try and correct the plaquette errors. If the second qubit 102b/third qubit 102c is flipped, then the errors in the third 104c and firstl 04a/second 104b and fourth 104d plaquettes are flipped and it only requires a single further flip, of the third qubit 102c/second qubit 102b, to revert the system back to the correct logical state.
  • first 106 and second 108 resonators are provided on interpenetrating square lattices within the qubit 102 array.
  • Each plaquette 104 is coupled to a single one of the first resonators 106 and each star is coupled to a single one of the second resonators 108.
  • the resonators induce a ranged interaction between the anyons (plaquette errors), which creates an energy cost associated with having spatially separated anyons (the separation between the anyons is measured from the centre of the respective plaquettes 104). Therefore, flipping the first spin 102a has a different energy cost to flipping the second 102b or third 102c spin, because flipping the central qubit 102a leaves the remaining anyons further apart, and so removes less energy from the system than flipping the second 102b or third 120c qubits.
  • a second primary repair qubit 1 14b is provided in each shadow qubit system 1 12, to address the error process illustrated by Figure 5A.
  • the second primary repair qubit 1 14b is tuned to be resonant with flipping the second 102b or third 102c qubits when there are four adjacent plaquette errors, and so the simpler route to returning the system to the correct logical state is favoured by the system 100.
  • Figure 5B shows the system 100 of Figure 1 where two consecutive errors have occurred.
  • the first error is in the first qubit 102a, as described in relation to Figure 2, causing plaquette errors in the first 104a and second 104b plaquettes.
  • the second error occurs in the third qubit 102c.
  • anyon pair separate.
  • plaquette errors anyons
  • a logical error can occur when the pair loop around the torus and annihilate at the opposite side to where they are created, forming a non-contractible loop upon annihilation.
  • the first 106 and second 108 resonators couple to the qubits such that processes that separate anyons have an associated energy cost.
  • the cost of separating an anyon pair is lower than creating a new anyon pair and varies as a function of the separation.
  • the rate at which anyon pairs separate is the same as the error rate, since further errors are required to cause the separation.
  • secondary repair qubits 1 14c-i are provided, resonant with the energies of various step wise anyon separations, to prevent separation of the anyon pair and to bring them back together.
  • FIG. 4 shows the energies of the seven secondary repair qubits 1 14c-i of the current embodiment as the locations of peaks.
  • Figure 6 shows a number of different configurations of secondary repair qubits 1 14c-i, as will be discussed in more detail below.
  • the energy in Figure 6 is divided by the coupling between the plaquette s/stars and the resonators (g), for ease of display.
  • the heights of the peaks in figure 6 correspond to the error correction rates induced by the secondary repair qubits.
  • each primary qubit 102 is the superconducting structure of a transmon qubit (C . Rigetti, S . Poletto, J. M.
  • the qubit structure can contain a trapped photon (state 1) or not (state 0).
  • state 1 a trapped photon
  • light can be used to control the qubits.
  • interactions with the outside environment can cause random errors, predominantly in the form of photon losses. Random errors can also be caused by other interactions, such as temperature fluctuations, or microscopic defects in the device.
  • the shadow lattice qubits 1 14 can also be formed in any suitable system.
  • the shadow lattice may be formed of the same system as the primary lattice.
  • the system may be formed in a stacked structure, with the photon trapping structure at the base, then the control circuitry, then the shadow lattice .
  • Qubits 102, 1 14 can decay, meaning they lose any information stored in them.
  • the primary qubits 102 are formed with as low a decay rate as possible, so that they can hold the same state for as long as possible, reducing the number of random errors caused.
  • the shadow lattice qubits 1 14 are formed as intentionally bad qubits, with a fast decay rate. This can be achieved in any number of suitable ways. For example, while the primary qubits 102 may be strongly electrically isolated, the shadow qubits 1 14 may only be weakly isolated.
  • the shadow qubits 1 14 may also be formed with a greater density of impurities, which increase the decay rate .
  • the structure of the shadow qubits 1 14 e.g.
  • the shadow qubits 1 14 may also be coupled to the primary qubits 102 in any suitable manner.
  • the shadow qubits 1 14 may be capacitively coupled through a dielectric or isolating layer, or may be coupled through vertical structures through the layer.
  • the coupling between the shadow qubits 1 14 and the primary qubits 102 may be switched on and off. This is particularly useful, if the coupling interferes with one or more processes in the operation of a quantum computer using the quantum logic system 100.
  • the lattice of first resonators 106 is implemented as a square grid of superconducting resonators, coupled capacitively to each other.
  • the lattice of second resonators 108 is also implemented as a square grid of superconducting resonators, coupled capacitively to each other.
  • the lattice of second resonators 108 is coupled to all the plaquettes 1 10, and the lattice of first resonators 106 to all the stars 104.
  • the strengths of these capacitive couplings can be identical between the two lattices, but the bare excitation energy (resonant frequency) of the first resonators 106 and the second resonators 108 should be substantially different.
  • the coupling between the qubits 102 in the plaquettes 104 and stars 108 can be implemented through a combination of a superconducting Josephson junction and a time dependent, oscillating magnetic flux bias, as described later on.
  • the oscillation frequency for the plaquette 104 and star 1 10 couplings should be chosen appropriately to match the energy of the resonators in the corresponding resonator lattice.
  • the overall rate at which errors are repaired is controlled by two factors: ( 1) the transfer of errors to the shadow qubits 1 14 and (2) the decay rate of the shadow qubits 1 14.
  • the shadow qubits 1 14 and structure of the system 100 should be controlled to maximise the repair rate and to ensure that it is higher than the error rate in the primary repair qubits 1 14. In one example, the repair rate should be ten times the error rate, or more.
  • passive error correction can significantly reduce the logical error rate of a quantum logic system 100.
  • the error rate can be further improved by providing an active error correction system (not shown), such as an algorithmic error correction system. Since the passive error correction has such a significant impact on the error rate, the requirements of the algorithmic system are significantly reduced as the algorithmic system is required to correct significantly fewer errors. This improves the overall performance of a quantum computer further.
  • the system 100 discussed above and illustrated in Figures 1 to 5 may have any number N of qubits 102. Since the N qubits correspond to the logical state, increasing N increases the number of successive random errors needed to generate a logical error, and so can increase the logical state lifetime . For some systems, increasing N allows more shadow qubits to participate in long-ranged error correction, improving the lifetime still further.
  • the shadow qubit systems 1 12 are provided with first 1 14a and second 1 14b primary repair qubits and secondary repair qubits 1 14c-i.
  • the second primary repair qubits 1 14b and/or the secondary repair qubits 1 14c-i may be omitted and some improvement in the lifetime still achieved.
  • the resonator 106, 108 may also be omitted.
  • the shadow lattice is formed of qubits 1 14.
  • any suitable quantum object may be used in the shadow lattice.
  • "bad" resonators with fast decay rates and energies tuned to the various transitions may be used.
  • the shadow lattice may be formed by a combination of shadow qubits 1 14 and shadow resonators.
  • the system of passive error correction could also be applied to non-continuous arrangements, if the edge effects are taken into account.
  • Systems with physical edges are attractive for the purposes of building a physical quantum computer.
  • edges are taken into account by truncating star terms into three-body interactions on top and bottom edges, and similarly truncating plaquette terms on left and right edges.
  • a system with edges has a twofold degenerate ground state, which can be parameterized by the eigenvalue of a string operation consisting of a chain of y basis operations connecting the top and bottom edges of the system; this operator commutes with H and returns opposite signs for the two ground states.
  • bit flips create or annihilate anyons in pairs, and the combination of the attractive ranged interaction and secondary repair qubits make it rare for anyons to wander apart before they are annihilated.
  • a y basis error occurring on the top or bottom edge is capable of creating a single anyon (at an energy cost V instead of 2V ).
  • x-basis errors occurring on the left or right edges also create single anyons.
  • the secondary repair qubits 1 14c-i act on pairs of anyons. Therefore, an isolated anyon can wander freely if it moves away from the edge before a primary repair qubit 1 14a corrects the error as there is no partner anyon remaining in the system. This can be compensated for by introducing an external "edge" potential acting on the anyons by locally adjusting V 0 (or a compensating "background charge” of further boson source terms) from plaquette to plaquette or star to star.
  • third primary repair qubits 1 14j are provided in the shadow qubit systems 1 12 at the edges.
  • the third primary repair qubits are tuned to have energies to eliminate single anyons, and the energies of first 1 14a and second 1 14b primary repair qubits are locally adjusted to take the changing values of V into account
  • Passive error correction through resonant energy transfer could also be applied to other quantum computing methods, such as adiabatic quantum computing.
  • Passive error correction has been modelled to illustrate its effectiveness. This modelling, and the results will now be discussed in more detail, and with reference to Figures 6 to 9.
  • the model is exemplary only, and simply meant to illustrate the effectiveness of passive error correction.
  • Figure 7 shows a system 200 made up of three primary qubits 202 arranged in a ring .
  • the three primary qubits 202 form the primary lattice and are coupled to each other through simple ferromagnetic Ising xx interactions 204.
  • Each primary qubit 202 is also coupled to a shadow qubit 206, through an exchange interaction 208
  • the Hamiltonian for the system 200 of Figure 2 (where P denotes primary qubits 202 and S denotes shadow qubits 206) is:
  • the shadow qubits 206 in their ground states in the z basis; and The three primary qubits 202 all in the - 1 state in the x basis (the first logical state of the system) or all in the + 1 state in the x basis (the
  • An error in a first qubit 202a creates an excitation of energy 4J, since all the primary qubits 202 are not in the same state (i.e. the system is not in a logical state). Due to the coupling of the primary qubits 202 to the shadow qubits 206, a single spin flip is not an eigenstate of the Hamiltonian and the excitation flops back and forth between the primary qubit 202a and the shadow qubit 206a. The shadow qubits 206 relax from an excited state to the ground state at a relaxation rate which is significantly larger than the rate at which errors occur in the primary qubits 202. Due to this relatively rapid decay, whenever the excitation occupies the shadow qubit 206a, it can decay and thus the shadow qubits 206 are nearly always in the ground state, creating an effective zero temperature bath for the primary system.
  • the resonance condition implies that to excite the shadow qubit 206a, the excitation must be removed from the primary qubit 202a. This gives single errors in the primary qubits an effective decay rate limited by the slower of the rate at which errors transfer to the shadow qubits 1 14a and the relaxation rate .
  • the repair of an error occurs in two steps: ( 1) reversible resonant transfer of an error to a shadow qubit 206 and (2) irreversible relaxation of the shadow qubit 206. It is the state of the primary qubits 202 that is important for logical states. Therefore, the effect of the shadow qubits 206 can be integrated out using Fermi's golden rule to generate the effective repair rate for eliminating errors in the primary qubits 202.
  • the total rate of a process which changes the energy of the primary lattice by an energy i.e . the effective repair rate
  • a spin flip is an operation in the y basis. Since the spin flip on the primary qubit 202a is removed through a second y basis operation, the combined process of a bit flip error and its subsequent correction does not measure the combined state of the primary qubits 202, and therefore cannot distinguish the two ground (logical) states.
  • the system 200 of Figure 7 is for illustration only.
  • if an error occurs on a primary qubit 202 in the z basis it will be corrected with a y basis operation by the shadow qubits 206, leading to a combined x basis operation that returns opposite signs for the two logical ground states.
  • the relative phase of an arbitrary superposition of the two logical ground states is not protected against some errors. Therefore, a system with topological order, as shown in Figures 1 to 5, where the logical ground states are not be distinguishable through local operations, should be used.
  • the Hamiltonian for the system 100 of Figures 1 to 5 consists of simple kinetic terms for propagating bosons, which are coupled to the qubits through driven plaquette 104 and star 1 10 couplings.
  • the plaquette 104 couplings are products of four x basis operators acting on all the spins in a plaquette 104: (4 )
  • the star 1 10 couplings are products of four y basis operators acting on all the spins in a star 1 10 :
  • the Hamiltonian can be diagonalised exactly through a similarity transformation (see G.D . Mahan, Many-Particle Physics (Springer, 2000) .
  • W is the gap parameter of the lattice
  • Hc Since U (the interaction potential between anyons) is positive definite, the ground state of H c is simply one where all W operators evaluate to + 1 (all Q operators evaluating to zero), and has a degeneracy controlled by the system's boundary conditions . As all the W operators commute with each other, and plaquette 102 or star 1 10 violations do not propagate, Hc can be simulated as a purely classical model .
  • the anyon pair do not propagate on their own . Since all the terms in the Hamiltonian (equation 10) commute with each other, an anyon (error) can only move from a plaquette 102 or star 1 10 to one of its neighbours through a further error (spin flip), induced by a coupling to an external system .
  • An error can be represented by a random operation in the y basis, which involve an energy transfer of (the bare excitation energy of the qubits; see appendix B) in the
  • the two values of ji correspond to the locations of the two plaquettes 104 which share the flipped spin. While the action the error itself is trivial (it merely flips the sign of the two plaquettes 104 which share that spin), the expansion of the exponential factor, is more complex.
  • the value of the exponential operator depends critically on the values of the two plaquette 104a, 104b operators which share the flipped qubit 102a.
  • the factor in equation 12 will renormalize the squared matrix elements of the shadow lattice couplings (see below), and for shadow lattice couplings with low enough energies, the shadow lattice will be unable to create or annihilate propagating bosons and only the renormalization factor need be considered when taking the bosons' dynamics into account.
  • primary quantum errors which have an energy-independent rate, can create or destroy bosonic excitations in the resonator lattices, as there is no resonance condition which suppresses the boson-generating terms in equation 1 1.
  • the bosons (A/B) have intrinsic loss rates and given the error rate, we can consider the propagating boson lattice to be empty of excitations whenever
  • Boson losses are harmless from a computational point of view; losing a boson of momentum k has the result:
  • the shadow lattice coupling is of the form:
  • the sites i are primary qubits 102 and the mn sites are shadow qubits 1 14.
  • the shadow lattice can be easily integrated out through a Fermi's golden rule calculation, yielding a set of transition rates for the primary lattice to incoherently transition from an initial state to a final state through a spin flip at site i. This rate is referred to as the repair function.
  • the initial and final states must be related by a single spin flip for the induced transition rate to be nonzero.
  • the total transition rate can be computed by simple summation, provided that the energy ranges covered by the groups 1 12 of shadow lattice qubits 1 14 are well separated from each other in comparison to the decay rate and the couplings.
  • the transition rate between two states due to a single spin flip thus depends primarily on the energy difference between the two states.
  • the fact that the shadow lattice couplings are comparatively weak enforces a resonance condition: to obtain an appreciable transition rate, there is at least one shadow lattice qubit 1 14 where the squared detuning is not
  • the transition rate is thus sharply peaked at This helps to ensure that the shadow lattice efficiently corrects errors without significantly inducing errors itself.
  • the total rate at which errors will occur on each primary qubit 102 is the total of the random error qubit error rate (independent of the system's 100 energetics) and the repair function for that qubit 102 (i) (dependent on system 100 energetics).
  • the first primary repair qubits 1 14a are chosen to be resonant with the excitation energy of a pair of defects plaquettes sharing a single spin, and have energy:
  • the coupling and decay rate are both chosen be small compared to the energy, so that the flip rate induced by this component of the shadow qubits 1 14 is negligible for any processes which do not correspond to repairing local pairs of defects. Steps which create or separate anyons occur due to the primary error rate (photon losses to the outside world, in a QED system), and thus have an energy independent rate.
  • the lifetime of the system 100 including first 1 14a and second 1 14b primary repair qubits scales as:
  • the second primary repair qubits 1 14b are chosen with energies near 2V, and lower coupling and decay rate than the first primary repair qubits.
  • the second primary repair qubits 1 14b can increase the average lifetime by a factor of 2 or more .
  • the secondary repair qubits 1 14c-i are chosen with a variety of energies. Each step separating an anyon pair increases the energy of the system 100, by an amount depending on the relative positions of the anyons.
  • j3 ⁇ 4E increase in energy of system due to separation of anyon pair by one step
  • the total lifetime can therefore be increased by a factor proportional to:
  • n number of separating steps
  • n ma x the average number of steps to separate anyons such that the potential is no longer strong enough to make the cost of separating them further significant.
  • n max can be quite large, meaning that reducing the primary error rate by a factor of a can reduce the logical error rate by ⁇ nmax , leading to an enormous increase in the quantum state lifetime .
  • n max of nearly 7 in some of our numerical simulations.
  • FIGS 8A, 8B, 8 C and 9 show numerical simulations of a qubit system 100 as shown in, and discussed with reference to, Figure 1 , to demonstrate the effectiveness of passive error correction.
  • Each primary qubit 102 is coupled to nine shadow qubits 1 14 as discussed in reference to Figure 4.
  • the precise value of the energy barrier is irrelevant so long as it is large compared to the motional energy scale g 2 /J and appropriately matched by the shadow lattice .
  • each data point is the average of 900 individual runs of the defect tracking algorithm discussed in appendix A.
  • Figures 8A-C illustrate the logical state lifetime for a toroidal system 100, of L by L plaquettes 104.
  • the lifetimes are given in the units of g 1 .
  • the ground state is actually four-fold degenerate, with degenerate states being mixed by a pair of anyons wrapping around either axis of the torus .
  • quantum error random y basis operations
  • shadow lattice parameters used for the passive error correction simulations are detailed in table 1 , which shows the parameters for the first 1 14a and second 1 14b primary repair qubits (first and second row) and the secondary repair qubits 1 14c-i (third to ninth row).
  • the repair function of the secondary repair qubits 1 14c-i is as shown in Figure 6.
  • the thick black line show the repair function for propagating boson gap of 0. 1
  • the thin dashed line shows the repair function for propagating boson gap of 0.2
  • the thin solid lines shows the repair function for propagating boson gap of 0.4.
  • Figure 8A illustrates the variation of the lifetime with lattice size for propagating boson gap of 0.4.
  • Figure 8B illustrates the variation of the lifetime with lattice size for propagating boson gap of 0.2.
  • Figure 8C illustrates the variation of the lifetime with lattice size for propagating boson gap 0. 1.
  • Figures 8A-C show that passive error correction can give logical state lifetimes several orders of magnitude larger than the lifetime of individual qubits (the error rate).
  • the best and worst average lifetimes determined by the simulation differ by a factor of 3 in L, 6 in qubit error rate but 10 4 in lifetime.
  • the logical state lifetime increases continuously with increasing lattice size L until a saturation point at L c is reached, due to the finite length scale / of the interaction potential U.
  • Figure 9 shows the effect of variation of the primary repair rate (i.e . the effect of the primary repair qubits 1 14a,b) . All curves in Figure 9 have boson gap of 0. 1 and qubit error rate of 10 ⁇ 4 g, and are the result of 400 simulations . The lowest curve (circles) of Figure 9 has the same primary repair rate as Figure 8C (and is therefore the same as the middle curve on Figure 8C) .
  • the squares illustrate the logical state lifetime with a fifty percent increase in the transfer rate to the qubit and the decay rate of the qubits, the diamonds a 100% increase and the triangles a 150% increase .
  • Figure 9 shows that doubling the primary repair rates can produce an improvement of more than a factor of twenty in the logical state lifetime for small lattices . This is because of multi-anyon processes, where the interactions between four or more anyons push the system's energetics off resonance with the shadow lattice and inhibit error correction or drag separated anyons further apart. Multi-anyon process are extremely significant contributions to the net logical error rate, since the instantaneous density of fluctuating anyon pairs (and their subsequent in influence on other anyons) will be reduced with an increase in primary error rate is doubled. This is particularly significant for systems 100 with a small gap parameter,
  • Figure 10 shows the logical state lifetime for a non-continuous system with edges, the system having L by L plaquettes 104. The lifetimes are again given in the units of g 1 . In the simulation, a linear "edge" potential gradient d was applied
  • the separate curves represent error rates of (from bottom curve to top curve) and a boson gap of 0 ⁇ was used Xable i again shows the appropriate parameters for the shadow qubits 1 14.
  • the third primary repair qubits were chosen with and .
  • N random numbers on the interval between 0 and 1 were generated, and when one of these numbers was less than the corresponding error probability, the spin was flipped. If the two plaquettes adjoining that spin were in the ground state (0), they were flipped, label them both by k and increment k by one so that the label is not reused later. If the spin sits on the top edge of the system (the periodic boundary, not a physical edge) k was appended to the edge list. Similarly, if one of the adjoining plaquettes was empty and the other contained an anyon, the values of the two plaquettes were exchanged, keeping track of the anyon's label p, and appending p to the edge list if the flipped spin lied on the edge.
  • both plaquettes adjoining the flipped spin contain anyons, the anyons were annihilated them and an error checked for. If the anyons had different labels p and q, these were removed and then all other instances of the label q (in the system and in the edge list) were changed to p. If the anyons shared the same label p, no relabelling was needed. If the flip occurred at the edge, label p was appended to the edge list. If the two anyons shared the same label before annihilation, the edge list was checked, and if p occurred an odd number of times, then the anyons were considered to have traversed a noncontractible loop and a logical error had occurred. The time t of the error was noted and the calculation aborted.
  • anyons could be created singly through a spin flip at the top or bottom edge; in each case a new label k was generated.
  • an appropriate relabelling was performed if needed (taking care to replace any instances of the replaced label in each edge list), and then check both edge lists, if the label p occurs in both edge lists, then an error had occurred.
  • An error in this case corresponds to a string which joins the two edges, and as there is no way to deform the string so that it couples to only one edge without creating or annihilating anyons, it is a topologically invariant operation.
  • Figure 1 1 shows a circuit 300a of 3 superconducting qubits 301 , 302, 303, coupled to a central resonator 306 through flux-biased Josephson junctions. All three qubits 301 , 302, 303 are charge-insensitive devices (such as transmon, flux or fluxonium qubits) and have excitation energies and the resonator 306 has excitation energy ⁇ R.
  • All three qubits 301 , 302, 303 are charge-insensitive devices (such as transmon, flux or fluxonium qubits) and have excitation energies and the resonator 306 has excitation energy ⁇ R.
  • the Josephson energy E'j of the couplings is small compared to the internal Josephson energy E of the qubits and compared to the excitation energy of the resonator. It is assumed that the nonlinearities of all three qubits 301 , 302, 303 are large compared to all other couplings, so that they can be treated as simple spin- 1/2 degrees of freedom. First 301 and second 302 qubits are coupled to each other through another small Josephson junction with energy E"j . The rest frame basis of the qubits is the z basis. An oscillating voltage is applied to the third qubit 303 through a capacitive coupling, and the third qubit 303 is coupled to the resonator 306 through two Josephson junctions with different flux biases. For charge noise-free superconducting qubits operated at a flux symmetry point, the operator identifications are and and and
  • the flux biases for the qubit-resonator and qubit-qubit couplings are:
  • the oscillating voltage applied to the third qubit 303 has the form Moving
  • Figure 12 illustrates the solutions to the Hamiltonian in equation 26 for an example construction, showing an eigensystem of the 3 -body gadget.
  • the circles plot the eigenvalues of ° of the eigenstates and the squares plot the energy in units of k.
  • the central qubit 307 and the two resonators 306a, 306b can be integrated out. With properly chosen counterterms, we arrive at:
  • J 3 is the coefficient of the three-body gadget.
  • Figure 14 illustrates the eigensystem of an example of this device .
  • the circles plot the eigenvalues of the eigenstates and the squares plot the energy in units of k.
  • a 5 -body gadget, as shown in Figure 15, could be constructed by chaining together three of the 3 -body gadgets, though this is not the simplest or most efficient way to generate a 5-body coupling,
  • the third qubit 303 of the three body gadget is replaced by a resonator. Additional terms could be added through the driven flux bias to eliminate couplings that are quadratic or higher order in the resonator's creation and annihilation operators a and i
  • the first 301 and second 302 qubits are then given

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Abstract

L'invention concerne un système logique quantique (100) configuré pour exécuter une correction d'erreur passive. Le système logique quantique (100) comprend une pluralité de qubits (102) formant une pluralité de groupes de qubits (104) et ayant un premier taux de décroissance, un circuit de commande (300) pour contraindre passivement l'état général de chaque groupe de qubits (104), et une pluralité d'objets quantiques auxiliaires (114) couplés par résonance aux groupes de qubits (104) de sorte que des erreurs dans l'un de la pluralité de qubits (102) sont transférées par résonance aux objets quantiques auxiliaires (114). Les objets quantiques auxiliaires (114) ont un second taux de décroissance, plus rapide que le premier taux de décroissance, de sorte que des erreurs de qubits transférées par résonance aux objets quantiques auxiliaires (114) sont plus susceptibles de décroître aux objets quantiques auxiliaires (114) plutôt qu'au qubit (102) à partir duquel elles sont transférées. Figure 4 à publier avec l'abrégé.
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WO2020081805A1 (fr) * 2018-10-17 2020-04-23 Rigetti & Co, Inc. Ressources quantiques divisées en lots
US10942804B2 (en) * 2018-03-23 2021-03-09 Massachusetts Institute Of Technology Physical-layer quantum error suppression for superconducting qubits in quantum computation and optimization
CN113949506A (zh) * 2020-07-17 2022-01-18 军事科学院系统工程研究院网络信息研究所 基于量子分发波形共享的安全通信方法

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US8022722B1 (en) * 2010-06-04 2011-09-20 Northrop Grumman Systems Corporation Quantum logic gates utilizing resonator mediated coupling
US8242799B2 (en) * 2010-11-16 2012-08-14 Northrop Grumman Systems Corporation System and method for phase error reduction in quantum systems

Cited By (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109428711A (zh) * 2017-08-23 2019-03-05 科大国盾量子技术股份有限公司 用于量子密钥分发系统的初始密钥纠错模块及方法
CN109428711B (zh) * 2017-08-23 2022-04-29 科大国盾量子技术股份有限公司 用于量子密钥分发系统的初始密钥纠错模块及方法
US10942804B2 (en) * 2018-03-23 2021-03-09 Massachusetts Institute Of Technology Physical-layer quantum error suppression for superconducting qubits in quantum computation and optimization
WO2020081805A1 (fr) * 2018-10-17 2020-04-23 Rigetti & Co, Inc. Ressources quantiques divisées en lots
CN113949506A (zh) * 2020-07-17 2022-01-18 军事科学院系统工程研究院网络信息研究所 基于量子分发波形共享的安全通信方法
CN113949506B (zh) * 2020-07-17 2023-09-19 军事科学院系统工程研究院网络信息研究所 基于量子分发波形共享的安全通信方法

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