arXiv:1105.4303 · Physical Review A

Quadratic dynamical decoupling with nonuniform error suppression

Gregory Quiroz, Daniel A. Lidar

Published Oct 17, 2011• 18 citations

Abstract

We analyze numerically the performance of the near-optimal quadratic dynamical decoupling (QDD) single-qubit decoherence errors suppression method [J. West et al., Phys. Rev. Lett. 104, 130501 (2010)]. The QDD sequence is formed by nesting two optimal Uhrig dynamical decoupling sequences for two orthogonal axes, comprising ${N}_{1}$ and ${N}_{2}$ pulses, respectively. Varying these numbers, we study the decoherence suppression properties of QDD directly by isolating the errors associated with each system basis operator present in the system-bath interaction Hamiltonian. Each individual error scales with the lowest order of the Dyson series, therefore immediately yielding the order of decoherence suppression. We show that the error suppression properties of QDD are dependent upon the parities of ${N}_{1}$ and ${N}_{2}$, and near-optimal performance is achieved for general single-qubit interactions when ${N}_{1}={N}_{2}$.

DOI: 10.1103/physreva.84.042328

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arXiv:1105.4303 · Physical Review A

Quadratic dynamical decoupling with nonuniform error suppression

Gregory Quiroz, Daniel A. Lidar

Published Oct 17, 2011• 18 citations

arXiv HTML PDF

Abstract

{N}_{1}

and

{N}_{2}

pulses, respectively. Varying these numbers, we study the decoherence suppression properties of QDD directly by isolating the errors associated with each system basis operator present in the system-bath interaction Hamiltonian. Each individual error scales with the lowest order of the Dyson series, therefore immediately yielding the order of decoherence suppression. We show that the error suppression properties of QDD are dependent upon the parities of

{N}_{1}

and

{N}_{2}

, and near-optimal performance is achieved for general single-qubit interactions when

{N}_{1}={N}_{2}

DOI: 10.1103/physreva.84.042328

Related papers

Quadratic dynamical decoupling: Universality proof and error analysis

Wan-Jung Kuo, Daniel A. Lidar

We prove the universality of the generalized QDD ${}_{{N}_{1}{N}_{2}}$ (quadratic dynamical decoupling) pulse sequence for near-optimal suppression of general single-qubit decoherence. Earlier work showed numerically that this dynamical decoupling sequence, which consists of an inner Uhrig DD (UDD) and outer UDD sequence using ${N}_{1}$ and ${N}_{2}$ pulses, respectively, can eliminate decoherence to $O({T}^{N})$ using $O({N}^{2})$ unequally spaced ``ideal'' (zero-width) pulses, where $T$ is the total evolution time and $N={N}_{1}={N}_{2}$ . A proof of the universality of QDD has been given for even ${N}_{1}$ . Here we give a general universality proof of QDD for arbitrary ${N}_{1}$ and ${N}_{2}$ . As in earlier proofs, our result holds for arbitrary bounded environments. Furthermore, we explore the single-axis (polarization) error suppression abilities of the inner and outer UDD sequences. We analyze both the single-axis QDD performance and how the overall performance of QDD depends on the single-axis errors. We identify various performance effects related to the parities and relative magnitudes of ${N}_{1}$ and ${N}_{2}$ . We prove that using QDD ${}_{{N}_{1}{N}_{2}}$ decoherence can always be eliminated to $O({T}^{\mathrm{min}{{N}_{1},{N}_{2}}})$ .We prove the universality of the generalized QDD${}_{{N}_{1}{N}_{2}}$ (quadratic dynamical decoupling) pulse sequence for near-optimal suppression of general single-qubit decoherence. Earlier work showed numerically that this dynamical decoupling sequence, which consists of an inner Uhrig DD (UDD) and outer UDD sequence using ${N}_{1}$ and ${N}_{2}$ pulses, respectively, can eliminate decoherence to $O({T}^{N})$ using $O({N}^{2})$ unequally spaced ``ideal'' (zero-width) pulses, where $T$ is the total evolution time and $N={N}_{1}={N}_{2}$. A proof of the universality of QDD has been given for even ${N}_{1}$. Here we give a general universality proof of QDD for arbitrary ${N}_{1}$ and ${N}_{2}$. As in earlier proofs, our result holds for arbitrary bounded environments. Furthermore, we explore the single-axis (polarization) error suppression abilities of the inner and outer UDD sequences. We analyze both the single-axis QDD performance and how the overall performance of QDD depends on the single-axis errors. We identify various performance effects related to the parities and relative magnitudes of ${N}_{1}$ and ${N}_{2}$. We prove that using QDD${}_{{N}_{1}{N}_{2}}$ decoherence can always be eliminated to $O({T}^{\mathrm{min}{{N}_{1},{N}_{2}}})$.

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We introduce a high-order dynamical decoupling (DD) scheme for arbitrary bounded system-bath interactions in the weak-coupling regime. Given any decoupling group $\mathcal G$ that averages the interaction to zero, our construction {guarantees the existence of pulse sequences with at most $(|\mathcal G|-1)K$ pulses}, while canceling all error terms linear in the system-bath coupling strength up to order $K$ in the total evolution time. As a corollary, for an $n$ -qubit system with $k$ -local system-bath interactions, we obtain an $\mathcal{O}(n^{k-1}K)$ -pulse sequence, a significant improvement over existing schemes with $\mathcal{O}(\exp(n))$ pulses (for $k=\mathcal{O}(1)$ ). The construction is obtained via a mapping to the continuous necklace-splitting problem, which asks how to cut a multi-colored interval into pieces that give each party the same share of every color. We provide explicit pulse sequences for suppressing general single-qubit decoherence, prove that the pulse count is asymptotically optimal, and verify the predicted error scaling in numerical simulations. For the same number of pulses, we observe that our sequences outperform the state-of-the-art Quadratic DD in the weak-coupling regime. We also construct explicit high-order sequences for suppressing representative 2-local noise models. Finally, the same construction extends to suppress slow, time-dependent classical noise and to filter-function design.We introduce a high-order dynamical decoupling (DD) scheme for arbitrary bounded system-bath interactions in the weak-coupling regime. Given any decoupling group $\mathcal G$ that averages the interaction to zero, our construction {guarantees the existence of pulse sequences with at most $(|\mathcal G|-1)K$ pulses}, while canceling all error terms linear in the system-bath coupling strength up to order $K$ in the total evolution time. As a corollary, for an $n$-qubit system with $k$-local system-bath interactions, we obtain an $\mathcal{O}(n^{k-1}K)$-pulse sequence, a significant improvement over existing schemes with $\mathcal{O}(\exp(n))$ pulses (for $k=\mathcal{O}(1)$). The construction is obtained via a mapping to the continuous necklace-splitting problem, which asks how to cut a multi-colored interval into pieces that give each party the same share of every color. We provide explicit pulse sequences for suppressing general single-qubit decoherence, prove that the pulse count is asymptotically optimal, and verify the predicted error scaling in numerical simulations. For the same number of pulses, we observe that our sequences outperform the state-of-the-art Quadratic DD in the weak-coupling regime. We also construct explicit high-order sequences for suppressing representative 2-local noise models. Finally, the same construction extends to suppress slow, time-dependent classical noise and to filter-function design.

quant-pharXiv:2602.05343Physical Review LettersFeb 6, 20260 citations

{N}_{1}

and

{N}_{2}

{N}_{1}

and

{N}_{2}

, and near-optimal performance is achieved for general single-qubit interactions when

{N}_{1}={N}_{2}

We prove the universality of the generalized QDD

{}_{{N}_{1}{N}_{2}}

(quadratic dynamical decoupling) pulse sequence for near-optimal suppression of general single-qubit decoherence. Earlier work showed numerically that this dynamical decoupling sequence, which consists of an inner Uhrig DD (UDD) and outer UDD sequence using

{N}_{1}

and

{N}_{2}

pulses, respectively, can eliminate decoherence to

O({T}^{N})

using

O({N}^{2})

unequally spaced ``ideal'' (zero-width) pulses, where

T

is the total evolution time and

N={N}_{1}={N}_{2}

. A proof of the universality of QDD has been given for even

{N}_{1}

. Here we give a general universality proof of QDD for arbitrary

{N}_{1}

and

{N}_{2}

. As in earlier proofs, our result holds for arbitrary bounded environments. Furthermore, we explore the single-axis (polarization) error suppression abilities of the inner and outer UDD sequences. We analyze both the single-axis QDD performance and how the overall performance of QDD depends on the single-axis errors. We identify various performance effects related to the parities and relative magnitudes of

{N}_{1}

and

{N}_{2}

. We prove that using QDD

{}_{{N}_{1}{N}_{2}}

decoherence can always be eliminated to

O({T}^{\mathrm{min}{{N}_{1},{N}_{2}}})

d

. It is expected that

d

scales with $\ensuremath{\tau}$, the total duration of the sequence, as ${\ensuremath{\tau}}^{\mathrm{min}{{N}_{x},{N}_{z}}+1}$, where

{N}_{x}

and

{N}_{z}

are the number of pulses of the outer and of the inner sequence, respectively. We show both numerically and analytically that the state of the bath can boost the performance of QDD under certain conditions: The scaling of QDD for a given number of pulses can be enhanced by a factor of 2 if the bath is prepared in a highly symmetric state and if the system Hamiltonian is SU(2) invariant.

We introduce a high-order dynamical decoupling (DD) scheme for arbitrary bounded system-bath interactions in the weak-coupling regime. Given any decoupling group

\mathcal G

that averages the interaction to zero, our construction {guarantees the existence of pulse sequences with at most

(|\mathcal G|-1)K

pulses}, while canceling all error terms linear in the system-bath coupling strength up to order

K

in the total evolution time. As a corollary, for an

n

-qubit system with

k

-local system-bath interactions, we obtain an

\mathcal{O}(n^{k-1}K)

-pulse sequence, a significant improvement over existing schemes with

\mathcal{O}(\exp(n))

pulses (for

k=\mathcal{O}(1)

). The construction is obtained via a mapping to the continuous necklace-splitting problem, which asks how to cut a multi-colored interval into pieces that give each party the same share of every color. We provide explicit pulse sequences for suppressing general single-qubit decoherence, prove that the pulse count is asymptotically optimal, and verify the predicted error scaling in numerical simulations. For the same number of pulses, we observe that our sequences outperform the state-of-the-art Quadratic DD in the weak-coupling regime. We also construct explicit high-order sequences for suppressing representative 2-local noise models. Finally, the same construction extends to suppress slow, time-dependent classical noise and to filter-function design.