Description
Quantum lattice code for a non-degenerate lattice, thereby admitting a finite-dimensional logical subspace. Codes on \(n\) modes can be constructed from lattices with \(2n\)-dimensional full-rank Gram matrices \(A\). Any GKP code can be generated from a Gram matrix in standard form via a Gaussian unitary transformation [3; Corr. 1].
The centralizer for the stabilizer group within the displacement operators case can be identified with the symplectic dual lattice \({\mathcal{L}}^{\perp}\) (i.e. all points in \(\mathbb{R}^{2n}\) that have integer symplectic inner product with all points in \({\mathcal{L}}\) ), such that logical operations are identified with the dual quotients \({\mathcal{L}}^{\perp}/{\mathcal{L}}\). The size of this dual quotient is the determinant of the Gram matrix, yielding the logical dimension \(d=\sqrt{\| \det{A}\|}\) [1]. Stabilizer generator matrices equivalent under symplectic transformations are classified by distinct Hermite normal forms [4].
The space of all single-mode GKP codes is the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed [5].
Protection
The level of protection against displacement errors is quantified by the Euclidean code distance \(\Delta=\min_{x\in {\mathcal{L}}^{\perp}\setminus {\mathcal{L}}} \|x\|_2\) [4]. There are upper bounds on this distance [4,6].Rate
Transmission schemes with multimode GKP codes achieve a lower bound on displacement noise and a lower bound on the thermal-noise Gaussian channel capacities [2,7–9]. Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel [2]. Particular families of GKP codes achieve the capacity of AD and amplification channels [10].Encoding
GKP codes with fixed \(n\) and prime-dimensional logical Hilbert space are symplectically related to a disjoint product of single-mode GKP codes on \(n\) modes, such that encoding via Gaussian unitaries is possible.Dissipative stabilization of finite-energy GKP states using stabilizers conjugated by cooling ([11], Appx. B) or damping operator, i.e., a damped exponential of the total occupation number [12,13].Gates
Gaussian operations and homodyne measurements on GKP states are classically simulable, and there is a sufficient condition for an additional element to achieve universal quantum computation [14]. There is an algorithm for GKP circuit simulation whose runtime scales with the amount of negativity of the Zak-Gross Wigner function [15].There is a relation between magic (i.e., how far away a state is from being a stabilizer state) and non-Gaussianity for GKP codewords [16,17]. In particular, implementing a non-Clifford logical gate requires a higher degree of non-Gaussianity than that expressed by ideal non-normalizable GKP states [17].By applying GKP error correction to Gaussian input states, computational universality can be achieved without additional non-Gaussian elements [18]. This procedure can be alternatively described as performing heterodyne detection on one half of a GKP encoded Bell pair. The cubic phase gate is not a suitable gate [19].Logical shadow tomography protocol [20].Decoding
Syndrome extraction is performed by measuring stabilizers and correcting. Issues arising from the use of finite-energy approximate states can be mitigated [21].The MLD decoder for Gaussian displacement errors is realized by evaluating a lattice theta function, and in general the decision can be approximated by either solving (approximating) the closest vector problem (CVP) [22] (a.k.a. closest lattice point problem) or by using other effective iterative schemes when, e.g., the lattice represents a concatenated GKP code [4,23–25]. While the decoder time scales exponentially with number of modes \(n\) generically, the time can be polynomial in \(n\) for certain codes [26].Babai's nearest plane algorithm [27] can be used for bounded-distance decoding [26].Combining AD noise with amplification yields displacement noise, the noise that GKP codes are designed to correct [9,28].ML decoder for correcting shift errors in GKP two-qubit gates [29].Fault Tolerance
Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [1]. For single-mode GKP codes, these operations correspond to non-trivial loops in the space of all single-mode GKP codes (the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed) [5]. Such gates provide another example of monodromy under the particular notion of parallel transport introduced in Ref. [30].Cousins
- Amplitude-damping (AD) code— Particular families of GKP codes achieve the capacity of AD and amplification channels [10].
- \(t\)-design— GKP states on \(n\) modes and their displaced versions for all possible lattices form a rigged 2-design for all \(n\) [34].
- Holographic code— GKP codespaces exist in the CFT dual of a particular holographic framework [35,36].
- Quadrature-amplitude modulation (QAM) code— Finite-energy GKP codes are quantum analogues of lattice-based QAM codes in that both use a subset of points on a lattice.
- Numerically optimized bosonic code— Numerically optimizing GKP code lattices yields codes for three, seven, and nine modes with larger distances and fidelities than known GKP codes [26]. Neural networks can be used to optimize approximate GKP states [37].
- Qutrit-Pauli group-representation code— The qutrit-Pauli group-representation code is a subcode of a two-mode GKP code [38].
Primary Hierarchy
Parents
GKP codes are \(n\)-mode quantum lattice codes with \(2n\) stabilizers, i.e., constructed using a non-degenerate lattice.
Gottesman-Kitaev-Preskill (GKP) code
Children
References
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Page edit log
- Jonathan Conrad (2022-07-05) — most recent
- Victor V. Albert (2022-07-05)
- Victor V. Albert (2022-03-24)
Cite as:
“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/multimodegkp