## Abstract

We demonstrate the protection of one bit of quantum information against all collective noise in three nuclear spins. Because no subspace of states offers this protection, the quantum bit was encoded in a proper noiseless subsystem. We therefore realize a general and efficient method for protecting quantum information. Robustness was verified for a full set of noise operators that do not distinguish the spins. Verification relied on the most complete exploration of engineered decoherence to date. The achieved fidelities show improved information storage for a large, noncommutative set of errors.

Quantum information is represented in terms of superposition states of elementary two-level systems, known as qubits. The coherence properties of such superpositions are essential to the extraordinary capabilities that quantum mechanics promises for quantum simulation (1), computation (2), and communication (3). At the same time, they are also extremely vulnerable to the decoherence processes that real-world quantum devices undergo due to unwanted couplings with their surrounding environment (4). Thus, achieving noise control is indispensable for practical quantum information processing (QIP). While a variety of strategies have been devised to meet this challenge, no single method can compensate for a completely arbitrary noise process. Rather, constructing a reliable QIP scheme depends crucially on the errors that happen. If the interaction with the environment is sufficiently weak, then, to a good approximation, a restricted set of errors dominates the information loss, and active quantum error correction (QEC) (5) can be successfully implemented. Another instance where the relevant errors are a subset of all possible errors occurs when the system-environment interaction, no matter how strong, exhibits a symmetry. This motivated passive noise control schemes based on encoding quantum information into “noiseless” (or “decoherence-free”, DF) subspaces (6–9). A DF subspace is spanned by states of the system that, up to a possible common phase factor, experience no evolution under the noise. In fact, the existence of such invariant states is only one of the possibilities by which the underlying symmetry may reflect into the evolution of the system, i.e., the one where all the properties defining the quantum state are effectively conserved. While this turns out to be an exceptionally favorable situation, symmetries always imply the existence of conserved quantities. The notion of a noiseless subsystem (NS) (10) captures this intuition, opening the way for exploiting symmetries in full generality. The basic insight is that protecting quantum information need not require protecting the entire state. A classical example is parity preservation under the action of paired bit-flips, where the sum (modulo 2) of a bit string is unchanged by flips of any two bits. Similarly, by encoding quantum information into the abstract subsystems corresponding to preserved degrees of freedom, noiselessness is guaranteed even if errors still evolve the overall system's state.

The fact that NSs cover the full range of possibilities for error-free storage has immediate practical significance: Not only can NSs exist in the absence of DF subspaces, as in the situation addressed by our experiment, but they can ensure the same degree of protection through more efficient encodings (10). Even in a scenario where memory resources are not a concern, NSs allow for a major conceptual advance in realizing noise control in QIP. The identification of appropriate NSs provides the key for linking passive stabilization schemes with active error control based on either QEC or quantum error suppression (11), resulting in a unified picture of noise control strategies that is not achievable in terms of subspaces alone (10–12). In particular, although a correspondence exists between degenerate QEC codes and DF subspaces (13), establishing a similar duality for arbitrary codes requires the more powerful NS formalism (10,12, 14). In fact, the deepest implication behind the NS idea is to point to the ultimate structure, a logical subsystem, where quantum information can reside. While the full impact of subsystems in QIP is still to be appreciated, this is likely to be substantially beyond noise protection itself—shedding light, for instance, on approaches to quantum universality (15) and entanglement (16).

A relevant symmetry arises when the environment couples to the qubits without distinguishing between them. This “far field” regime, where each qubit “feels” the same fluctuation, is entered whenever the spatial separations between the qubits are small relative to the correlation length of the environment. The resulting “collective” noise behavior provides the paradigmatic situation for discussing passive noise control. Experimental efforts to date have been limited to demonstrating the existence of protected subspaces under special types of collective noise—a single DF state of two photons (17) and a DF subspace of two trapped ions (18). For arbitrary collective noise acting on three qubits there are no protected subspaces, but there is a NS. Here, we realize this minimal, three-qubit NS to guarantee the preservation of one qubit against the most general collective noise.

Our physical system is composed of three spin 1/2 particles. Let σ_{α}
^{(j)}, α = *x*,*y*, *z*, denote Pauli spin operators acting on the*j*th qubit, *j* = 1, 2, 3. Under collective noise conditions, only global angular momentum operators*S*
_{α} = (σ_{α}
^{(1)} + σ_{α}
^{(2)} + σ_{α}
^{(3)})/2 contribute to the system-environment interaction. If the spins are initially in a pure state ϱ_{in}, the effect of the environment may be depicted in terms of a quantum operation that leaves them in a mixed state, ϱ_{in} → ϱ_{out} = ℰ(ϱ_{in}) = Σ_{a}
*E _{a}
* ϱ

_{in}

*E*

_{a}

^{†}, for a set of error operators {

*E*} satisfying Σ

_{a}_{a}

*E*

_{a}

^{†}

*E*= . The possible errors that the error generators

_{a}*S*

_{α}can induce belong to the “interaction algebra”

**A**(10), which contains all the linear combinations of arbitrary products of

*S*

_{α}and the identity. In the weak-noise limit addressed by ordinary QEC, the relevant first-order errors essentially coincide with the generators

*S*

_{α}. Error operators involving products of any number of

*S*

_{α}occur for sufficiently strong or sufficiently long exposure to noise. Thus,

**A**provides the appropriate tool for discussing protection of information with “infinite-distance,” against arbitrarily large noise strength and/or order in time. The interaction algebra

**A**

_{c}for general collective noise consists of all the totally symmetric operators—which reflects the permutation-symmetry of the noise. For three qubits, the dimension of the symmetric operators' subspace is 20, so an error basis for describing an arbitrary collective noise process can be constructed from 20 three-spin operators (out of the possible 64). Simpler collective noise models, like the ones probed in (17, 18), correspond to smaller error algebras. For instance, collective dephasing with arbitrary strength, ℰ

_{z}, is described by an abelian subalgebra

**A**

_{z}spanned by four elements, ,

*S*,

_{z}*S*

_{z}

^{2},

*S*

_{z}

^{3}—and similarly for any fixed axis. The full, nonabelian

**A**

_{c}can be induced by cascading noise processes along at least two noncommuting spatial directions, e.g., ϱ

_{out}= ℰ

_{zx}(ϱ

_{in}) = ℰ

_{z}(ℰ

_{x}(ϱ

_{in})).

The NS reported here lives in the four-dimensional subspace ℋ_{1/2} of states carrying total angular momentum*S* = 1/2 (10, 19). Basis states for ℋ_{1/2} are specified by two quantum numbers, |λ, *s _{z}
*〉, where

*s*= ± 1/2 is the eigenvalue of

_{z}*S*, and λ = 0,1 accounts for the existence of two distinct pathways leading to total angular momentum

_{z}*S*= 1/2 (14, 19). Because each of the error generators

*S*

_{α}acts equivalently and nontrivially only within each path, the quantum number λ is preserved under the action of arbitrary errors in

**A**

_{c}. Thus, the logical subsystem

*L*supported by λ is a NS under general collective noise. In other words, ℋ

_{1/2}can be pictured as the state space of two abstract qubits:

*L*, which is fully protected against collective errors, and

*Z*, which carries all the entropy inserted by the noise. The stability of information encoded in

*L*against all error operators in

**A**

_{c}characterizes

*L*as an infinite-distance QEC code for collective noise (10). Because four physical qubits are required for attaining the same goal through a DF subspace (6), the three-qubit NS realizes the smallest one-bit noiseless quantum memory under

**A**

_{c}.

In terms of the single-qubit input ρ_{in} and output ρ_{out} for the data spin alone (see Fig. 1 for outline of experiment), the overall effect of the procedure is described by a one-bit quantum operation that maps ρ_{in} → ρ_{out} = ε(ρ_{in}). Ideally, ρ_{out} = ρ_{in}. In the presence of unavoidable imperfections, we invoke the entanglement fidelity*F*
_{e} (20) as the appropriate measure for quantifying the preservation of the quantum data. For a given process ε,*F*
_{e}(ε) = 1 if and only if ε perfectly preserves every input state. A complete characterization of ε from experimentally available data can be obtained by “quantum process tomography” (21), which relies on measuring the output states generated from a complete set of independent input states. Let |0〉 and |1〉 denote the eigenstates of σ_{z}, and define |+〉 = (|0〉 + |1〉)/ , |+*i*〉 = (|0〉 + *i* |1〉)/, respectively. Under the assumption that ε describes a “unital” process for which ε( ) = ,*F*
_{e}(ε) can be calculated as (22)where*F*
_{|ψin}
_{〉} =*Tr*{|ψ_{in}〉〈 ψ_{in}|ε(|ψ_{in}〉〈ψ_{in}|)} is the input-output fidelity for the intended one-bit pure input state |ψ_{in}〉.

Our implementation was performed by liquid-state nuclear magnetic resonance (NMR) on a sample of ^{13}C-labeled alanine (Fig. 2) in D_{2}O solution, with a 300-MHz Bruker Avance spectrometer. NMR QIP has been extensively discussed in the literature (23). Room-temperature NMR qubits exist in highly mixed, separable states. Thus, NMR QIP relies on “pseudo-pure” (p.p.) states whose traceless (or “deviation”) component is proportional to that of the corresponding pure state. The identity component of the density matrix is unobservable and can be treated as a constant under the assumption of unital dynamics (24). Initialization of the qubits in one of the 3-spin p.p. input states ϱ _{in}
^{p.p.} = |00ψ_{in}〉〈00ψ_{in}|, where the data-bit state |ψ_{in}〉 = |0〉, |+〉, |+*i*〉 is defined above, was accomplished with standard gradient-pulse techniques. A sequence of transformations generating the above states from the 3-spin thermal equilibrium state, as well as other implementation details, can be found in (25). State preparation was verified by tomographically reconstructing the resulting 3-spin deviation density matrix. An amount of identity component able to optimize the fidelity with the intended 3-spin p.p. state was added to the experimental ϱ _{in}
^{p.p.}(26) and maintained throughout the analysis. The evolution of a p.p. state is equivalent to the corresponding pure-state evolution under unital dynamics. The logical manipulations involved in the NS encoding and decoding were mapped into ideal pulse sequences by standard quantum network methods (23). Pulses were implemented by modulating the internal Hamiltonian of the alanine molecule (Fig. 2) with externally controlled radio-frequency (RF) magnetic fields.

To investigate the NS performance in a controlled way, the time delay between encoding and decoding is designed to implement a net evolution of the spins under a desired collective noise model. Unwanted, symmetry-breaking dynamics generated by the molecule's internal Hamiltonian were refocused by using standard average Hamiltonian techniques (23). A variety of collective noise processes can be engineered through gradient-diffusion methods (27, 28). A pulsed magnetic field gradient ∂*B _{z}
*/∂

*z*parallel to the static quantizing field

*B*causes the spins to precess with a

_{z}*z*-dependent Zeeman rate, thereby acquiring a phase factor that is identical for each species but varies linearly with

*z*(29). These spatially dependent phases add incoherently to zero when the integrated signal from the sample is measured. Thus, the action of a strong gradient pulse over the ensemble amounts to an incoherent implementation of all the possible collective phase errors, emulating the error algebra

**A**

_{z}of the strong dephasing regime. Collective noise along an arbitrary axis is induced by sandwiching a

*z*-noise between RF pulses, effecting a collective rotation of the spin state to the desired axis (28). The incoherent action of a single gradient pulse could be refocused by a second inverse gradient. A truly irreversible, decoherent implementation of collective noise is obtained by allowing for molecular diffusion to take place for a time Δ

*t*before applying an inverse gradient. Because the molecules have moved, the spins' phases are not returned to their original values but are randomly modified. Thus, the combined gradient-diffusion action results in an exponential signal loss whose effective decay rate 1/τ is proportional to the diffusion constant and tunable with the gradient intensity (27).

Decoherent collective noise of variable strength was engineered by stepping the gradient amplitude from 0 to the maximum achievable value of about 0.5 T/m, and applied for a fixed evolution time*t*
_{ev}. All noise strengths 1/τ were calibrated by independent measurements with a gradient stimulated echo sequence (27). The attained ratios*t*
_{ev}/τ are sufficient to push the evolution beyond the weak-dephasing regime that can be compensated for by QEC (30). Separate experiments were performed to expose the NS-encoded qubit and the unencoded data spin,*C*
_{3}, to single-axis collective *z* and*y* noise (Fig. 3). Both the unencoded and NS-encoded data are fit to a decaying exponential model,*F*
_{e} = *A*
_{1}exp(−*t*
_{ev}/τ) + *B*(31). The asymptotic *F*
_{e} value given by *B* is a relevant figure of merit for lower-bounding the entanglement fidelity attainable for single-axis noise of arbitrary strength. The test data are expected to decay under a single-qubit dephasing channel along *y*—i.e.,*A*
_{1} = *B* = 0.50—as opposed to the observed values of *A*
_{1} = 0.51 ± 0.04, *B* = 0.43 ± 0.03. For the NS data, the theory predicts a constant unit *F*
_{e}. In the experiment, both a constant term, *B* = 0.64 ± 0.02 (*y*) and *B* = 0.62 ± 0.02 (*z*), and a small decaying contribution,*A*
_{1} = 0.03 ± 0.03 (both *y*and *z*), are seen. For both the NS- and the unencoded cases, departures from the ideal values are explained by pulse imperfections and by natural relaxation processes, whose action is assumed to be independent of the applied-noise strength. Several remarks are in order concerning the NS data. First, the fact that *A*
_{1}is compatible with zero suggests that the measured signal predominantly originates from the NS, as a signal from other locations would decay with increasing applied noise. Second, because *B* well exceeds the threshold value of 0.50, the implementation guarantees, in principle, entanglement preservation for arbitrary noise strength (32). Finally, the fidelities achieved for sufficiently strong noise imply an actual improvement in preserving quantum information via the NS code.

A variety of incoherent collective error processes were also implemented to explore exhaustively the NS robustness under strong single- and multiple-axes noise that fully probe the nonabelian error algebra of general collective noise. Incoherent implementations have the advantage that full-strength error models can be induced quickly as compared with the natural relaxation time scales. The experimental data for both unencoded and NS-encoded evolutions are summarized in Table 1. As in the decoherent case, the*F*
_{e} values for single-axis noise, ℰ_{α}, α = *x*, *y*,*z*, demonstrate the infinite error-correcting behavior of the NS code against errors in the corresponding abelian subalgebra **A**_{α}. Robustness against the full **A**_{c} is verified through the composite processes (ε_{zx}, ε_{zy}, ε_{yzx}) obtained by sequentially implementing evolution periods corresponding to single-axis error models along different directions. The measured input-output and entanglement fidelities are consistent with the expectation that single-axis and composite noise processes induce full phase-damping and full depolarization on the unencoded data bit—with predicted fidelities of 0.50 and 0.25, respectively. Infinite-distance error-correcting behavior with respect to **A**_{c} is established by the unchanged fidelity levels observed in the presence of the applied noise relative to the corresponding no-error case. The data show a substantial increase in the amount of information retained under the action of the applied error models.

NSs provide extremely efficient means of preserving quantum information whenever a dominant symmetry occurs in the noise. Therefore, symmetry should be a leading criterion for engineering QIP. We have shown that the implementation of NSs is within the reach of current quantum information technologies. Although the attained fidelities are less than ideal and point to the need for improved quantum control capabilities, our implementation convincingly demonstrates improvement in error-correcting a class of both abelian and nonabelian error models via an infinite-distance one-qubit quantum code. The encoding, decoding, and verification techniques used here are exportable to other quantum information devices, where the dominant noise mechanisms are collective in nature. Besides providing the chief source of ambient noise in trapped ion QIP (18), collective noise behavior is also expected to be predominant in solid-state architectures based on either nuclear spin (33) or quantum dot (34) arrays. NS coding, possibly combined with fault-tolerant and active control methods, can thus play a practical role for both reliable storage and manipulation of quantum information in future QIP.