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Quantum computations on a topologically encoded qubit

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Science  18 Jul 2014:
Vol. 345, Issue 6194, pp. 302-305
DOI: 10.1126/science.1253742

Fault-tolerant quantum computing

Quantum states can be delicate. Attempts to process and manipulate quantum states can destroy the encoded information. Nigg et al. encoded the quantum state of a single qubit (in this case, a trapped ion) over the global properties of a series of trapped ions. These so-called stabilizers protected the information against noise sources that can degrade the single qubit. The protocol provides a route to fault-tolerant quantum computing.

Science, this issue p. 302

Abstract

The construction of a quantum computer remains a fundamental scientific and technological challenge because of the influence of unavoidable noise. Quantum states and operations can be protected from errors through the use of protocols for quantum computing with faulty components. We present a quantum error-correcting code in which one qubit is encoded in entangled states distributed over seven trapped-ion qubits. The code can detect one bit flip error, one phase flip error, or a combined error of both, regardless on which of the qubits they occur. We applied sequences of gate operations on the encoded qubit to explore its computational capabilities. This seven-qubit code represents a fully functional instance of a topologically encoded qubit, or color code, and opens a route toward fault-tolerant quantum computing.

A fully fledged quantum computer can be used to efficiently solve notoriously difficult problems, such as factoring large numbers or simulating the dynamics of many-body quantum systems (1). Technological progress has enabled the implementation of small-scale prototype quantum computing devices on diverse physical platforms (2). Sophisticated fault-tolerant quantum computing (FTQC) techniques have been developed for the systematic correction of errors that dynamically occur during storage and manipulation of quantum information (35). For quantum error correction, Calderbank-Shor-Steane (CSS) codes (4, 5) enable independent detection and correction of bit and phase flip errors, as well as combinations thereof. Furthermore, quantum information processing is substantially facilitated in quantum codes in which logical operations on encoded qubits are realized by the bitwise application of the corresponding operations to the underlying physical qubits (i.e., in a transversal way). This property prevents uncontrolled propagation of errors through the quantum hardware, which in turn is essential to enter the FTQC regime (1). Ultimately, reliable quantum memories and arbitrarily long quantum computations are predicted to become feasible for appropriately designed quantum codes, once all elementary operations are realized in a fault-tolerant way and with sufficiently low error rates (6, 7).

To date, topological quantum computing represents the most promising and realistic approach toward FTQC. In this method, the encoding of quantum information in global properties of a many-particle system provides protection against noise sources that act locally on individual or small sets of qubits (8). Most prominently, topological quantum computing offers highly competitive error thresholds as high as 1% per operation (912), which is within reach of current experimental capabilities (1315) and typically about two orders of magnitude larger than in schemes using concatenated quantum codes (7).

Within topological quantum computing, topological color codes (16, 17) offer the distinctive feature that the entire group of Clifford gate operations, allowing for arbitrary 90° and 180° qubit rotations, can be implemented transversally (1). This versatile set of operations directly enables protocols for quantum distillation of entanglement, quantum teleportation, and dense coding with topological protection (16). Moreover, a universal gate set, enabling the implementation of arbitrary quantum algorithms, can be achieved by complementing the Clifford operations with a single non-Clifford gate (1). For color codes in two-dimensional (2D) architectures (16), such an additional gate can be realized by a technique known as magic-state injection (18). Remarkably, this method is not needed in 3D color codes that enable implementation of a universal gate set using exclusively transversal operations (17).

Previous experiments have demonstrated the correction of a single type of error by the three-qubit repetition code (1921), correction of bit and phase flip errors by the non–CSS-type five-qubit code in nuclear magnetic resonance systems (22, 23), as well as elements of topological error correction in the framework of measurement-based quantum computation (24). Here, we demonstrate a quantum error-correcting seven-qubit CSS code (5), which is equivalent to the smallest instance of a 2D topological color code (16). The application of multiple operations from the entire, transversally implemented set of logical single-qubit Clifford gates represents the realization of quantum computations on a fully correctable encoded qubit.

Two-dimensional color codes are topological quantum error-correcting codes that are constructed on underlying 2D lattices (16) for which three links meet at each vertex and three different colors are sufficient to assign color to all polygons (plaquettes) of the lattice, such that no adjacent plaquettes sharing a link are of the same color. The smallest fully functional 2D color code involves seven qubits (Fig. 1A) and consists of a triangular, planar code structure formed by three adjoined plaquettes with one physical qubit placed at each vertex. Color codes are stabilizer quantum codes (1, 25), which are defined by a set of commuting, so-called stabilizer operators {Si}, each having eigenvalues +1 or –1. More precisely, the code space hosting logical or encoded quantum states Embedded Image is fixed as the simultaneous eigenspace of eigenvalue +1 of all stabilizers, Embedded Image. In color codes, there are two stabilizer operators associated with each plaquette, which for the seven-qubit color code (Fig. 1A) results in the set of four-qubit X and Z-type operators Embedded Image (1)where Xi, Yi, and Zi denote the standard Pauli matrices acting on the ith physical qubit with the computational basis states Embedded Image and Embedded Image (1). The stabilizers in Eq. 1 impose six independent constraints on the seven physical qubits and thus define a two-dimensional code space, which allows the encoding of one logical qubit. The logical basis states Embedded Image and Embedded Image spanning the code space are entangled seven-qubit states and are given as the eigenstates of the logical operator ZL = Z1Z2Z3Z4Z5Z6Z7, where Embedded Image and Embedded Image (16, 26).

Fig. 1 The topologically encoded qubit: its physical implementation and initialization.

(A) One logical qubit is embedded in seven physical qubits forming a 2D triangular planar code structure of three plaquettes. The code space is defined via six stabilizer operators Embedded Image and Embedded Image, each acting on a plaquette that involves four physical qubits. (B) For the physical realization, qubits are encoded in electronic states of a linear string of ions. These ion qubits can be manipulated via laser interactions that realize a universal gate set consisting of single-ion phase shifts, collective operations, and a collective entangling gate, which is complemented by a single-ion spectroscopic decoupling and recoupling technique (25). (C) Encoding of the logical qubit is achieved by coherently mapping the input state Embedded Image onto the logical state Embedded Image, using a quantum circuit that combines plaquette-wise entangling operations with decoupling and recoupling pulses (yellow and white squares, respectively) (26). Dashed lines denote decoupled or inactive qubits; solid lines denote recoupled or active qubits.

For the physical realization of a topologically encoded qubit, we store seven 40Ca+ ions in a linear Paul trap. Each ion hosts a physical qubit, which is encoded in (meta)stable electronic states (27). Within our setup (Fig. 1B), we realize a high-fidelity universal set of quantum operations consisting of single-ion phase shifts, collective rotations, and a collective entangling gate. Additionally, a single-ion spectroscopic decoupling technique enables the collective entangling operation to act only on subsets of qubits (26).

The initial preparation of the logical state Embedded Image (encoding) is realized deterministically by the quantum circuit shown in Fig. 1C. First, the seven-ion system is prepared in the product state Embedded Image, which satisfies the required three Embedded Image and Embedded Image conditions. In three subsequent steps, plaquette-wise entangling operations are applied to also satisfy the three Embedded Image constraints. For each step, three of the seven physical qubits are spectroscopically decoupled prior to the application of the collective entangling gate. Subsequently, Greenberger-Horne-Zeilinger–like entanglement between the four qubits belonging to one plaquette of the code is created with a fidelity of 88.8(5)% (where the value in parentheses is the error in the last significant digit). The entire encoding sequence involves three collective entangling gates and 108 local single-qubit rotations (26). The quantum state fidelity of 32.7(8)% of the system in state Embedded Image surpasses the threshold value of 25% (by more than 9 standard deviations), above which genuine six-qubit entanglement is witnessed, thereby clearly indicating the mutual entanglement of all three plaquettes of the code (26).

Two factors govern the quality of the created logical state Embedded Image (Fig. 2A): (i) its overlap with the code space, and (ii) its accordance within the code space with the target encoded state, which is related to the expectation value of the logical operator ZL. Residual populations outside the code space are indicated by deviations of the six stabilizer expectation values, in our case on average 0.48(2), from the ideal value of +1. A more detailed analysis shows that within the code space, the fidelity between experimental and target state is as high as 95(2)%, whereas the expectation value of Embedded Image and the overall fidelity between the experimentally realized and the ideal state Embedded Image are currently limited by the overlap with the code space of 34(1)% (26).

Fig. 2 Effect of arbitrary single-qubit errors on the encoded logical qubit.

(A) The initial logical state Embedded Image, prior to the occurrence of single-qubit errors, is reflected (i) by the error syndrome, in which all six Embedded Image and Embedded Image stabilizers are positive-valued, and (ii) by a positive (vanishing) expectation value of the logical operator ZL (XL). (B) A bit flip error (red wiggled arrow) on qubit 2 (marked in black) affects the blue and red plaquettes (visualized by gray-shaded circles) and manifests itself by negative Embedded Image and Embedded Image expectation values and a ZL sign flip. (C) A Z5 phase flip error only affects the blue plaquette and results in a sign flip of Embedded Image. (D) A Y3 error—equivalent to a combined X3 and Z3 error—affects all three plaquettes and induces a sign change in all six stabilizers and ZL. Double-error events, such as a Z5 phase flip (Fig. 2C), followed by a Z2 (E) or a Z3 error (F) result in an incorrect assignment of physical errors, as the detected stabilizer patterns are indistinguishable from single-error syndromes—here, the ones induced by a Z1 (Fig. 2E) or a Z4 (Fig. 2F) error (26). In the correction process, this eventually results in a logical error—here, a ZL phase flip error. Stabilizer violations can under subsequent errors hop (white non-wiggled arrow) to an adjacent plaquette, as in Fig. 2E, where the violation disappears (open gray circle) from the blue and reappears on the red plaquette. Alternatively (Fig. 2F), they can disappear (from the blue plaquette), split up (white branched arrow), and reappear on two neighboring plaquettes (red and green). This rich dynamical behavior of stabilizer violations is a characteristic signature of the topological order in color codes (16, 26).

It is a hallmark feature of topologically ordered states that these cannot be characterized by local order parameters, but only reveal their topological quantum order in global system properties (8, 26). We experimentally confirmed this intriguing characteristic for the topologically encoded seven-qubit system in state Embedded Image by measuring all subsets of reduced two-qubit density matrices, which yielded an average Uhlmann fidelity of 98.3(2)% with the two-qubit completely mixed state, showing the absence of any single- and two-qubit correlations. In contrast, the presence of global quantum order is signaled for the system size at hand by nonvanishing three-qubit correlations Embedded Image (26).

We next consider the error correction properties of the encoded qubit. Single-qubit errors lead the system out of the logical code space and manifest themselves as stabilizer eigenstates of eigenvalue –1 (stabilizer violations) associated to one or several plaquettes. Using the available gate set, we coherently induce all single-qubit errors on the encoded state Embedded Image and record the induced error syndromes provided by the characteristic pattern of six stabilizer expectation values (26) (see Fig. 2 for a selection). The experimental data reveal the CSS character of the quantum code: Starting in Embedded Image (Fig. 2A), single-qubit X errors manifest themselves as violations of Z-type stabilizers only, whereas single-qubit Z errors manifest themselves as violations of X-type stabilizers only (Fig. 2, B and C); the effect of single-qubit Y errors is equivalent to a combined X and Z error and is signaled by the simultaneous violation of the corresponding X- and Z-type stabilizers (Fig. 2D). For our experimental statistical uncertainties, the measured characteristic error syndromes can be perfectly assigned to the underlying induced single-qubit errors (26). Shown in Fig. 2, E and F, are data in which the code has been exposed to two single-qubit errors, whose correction exceeds the capabilities of the seven-qubit code (5, 16).

In topological color codes, quantum information is processed by logical gate operations acting directly within the code space (16, 17). The entire group of logical Clifford gates generated by the elementary gate operations Z, X, the Hadamard H, and the phase gate K,Embedded Image (2)as well as the C-NOT gate for registers containing several logical qubits, can be realized transversally. We implement the logical Clifford gates ZL = Z1Z2Z3Z4Z5Z6Z7, XL = X1X2X3X4X5X6X7, HL = H1H2H3H4H5H6H7, and KL = K1K2K3K4K5K6K7 by local operations that are realized by a combination of single-ion and collective rotations (26) (Fig. 1B). After initializing the logical qubit in the state Embedded Image, we prepare all six eigenstates of the logical operators XL, YL, and ZL, which requires quantum circuits consisting of up to three elementary logical Clifford gate operations (Fig. 3, A and B). The encoded qubit evolves as expected under the logical gate operations, as indicated by the characteristic changes in the pattern of XL, YL, and ZL, expectation values. This result is corroborated by quantum state fidelities within the code space of {95(2), 85(3), 87(2)}% of the experimental logical states Embedded Image with the expected ideal states, which indicates a high performance of the transversal (i.e., bitwise) logical Clifford gate operations (26).

Fig. 3 Single-qubit Clifford gate operations applied on a logical encoded qubit.

Starting from the logical Embedded Image state, sequences of logical Clifford gate operations {XL, HL} in (A) and {HL, KL, XL} in (B) are applied consecutively in a transversal way (i.e., bit-wise) to realize all six cardinal states Embedded Image of the logical space of the topologically encoded qubit. The dynamics under the applied gate operations is illustrated by rotations of the Bloch vector (red arrow) on the logical Bloch sphere as well as by the circuit diagram in the background. Each of the created logical states is characterized by the measured pattern of Embedded Image and Embedded Image stabilizers and the logical Bloch vector, with the three components given by the expectation values of the logical operators XL, YL, and ZL. The orientation of the logical Bloch vector changes as expected under the logical gate operations.

We further explore the computational capabilities of the encoded qubit by executing a longer encoded quantum computation, which consists of up to 10 logical XL gate operations applied to the system initially prepared in Embedded Image (Fig. 4A). Here, the logical qubit flips as expected between the YL eigenstates, as witnessed by alternating (vanishing) expectation values of the logical YL (XL, ZL) operator (Fig. 4B), accompanied by a moderate decay of the average stabilizer expectation values of only 3.8(5)% per logical gate operation, which is in quantitative agreement with the fidelities of our single-ion and collective local operations (26). Because the duration of 10 XL gates is more than one order of magnitude shorter than the measured coherence time of encoded logical states (26), the performance of the encoded quantum computation of Fig. 4B is currently not limited by the lifetime of the logical states but is predominantly determined by imperfections of the logical Clifford gate operations.

Fig. 4 Repetitive application of logical quantum gate operations.

(A) Preparation of the Embedded Image state by applying a HL, KL, and XL gate operation on the qubit initially prepared in the Embedded Image state. (B) Subsequently, flips between the logical Embedded Image and Embedded Image states are induced by consecutively applying logical XL gate operations up to 10 times. The sign flip of the YL expectation value (red diamonds) after each step clearly signals the induced flips of the logical Bloch vector, whereas the expectation values of ZL (blue squares) and XL (black circles) are close to zero, as expected [the average of Embedded Image yields {0.01(1), –0.01(1)}] (26). Average Embedded Image stabilizer expectation values after each XL gate are shown as gray bars; average Embedded Image stabilizer expectation values after each XL gate are shown as green bars.

Future objectives aiming at an extension of the demonstrated quantum information processing capabilities include the use of one additional (unprotected) ancillary qubit for the implementation of a non-Clifford gate, toward universal encoded quantum computation (1, 18). Furthermore, for higher experimental fidelities, repetitive application of complete error correction cycles is achievable (26) by incorporating previously demonstrated measurement and feedback techniques (28) for quantum nondemolition readout of stabilizers via ancillary qubits; this would represent progress toward the goal of keeping an encoded qubit alive. The demonstrated concepts can be adapted to 2D ion-trap arrays (29) as well as other scalable architectures including optical, atomic, molecular, and solid-state systems (2). The continuing technological development of these platforms promises fault-tolerant operating of larger numbers of error-resistant, topologically protected qubits, and thereby marks a clear, although technologically challenging, path toward the realization of FTQC.

Supplementary Materials

www.sciencemag.org/content/345/6194/302/suppl/DC1

Supplementary Text

Figs. S1 to S7

Tables S1 and S2

Supplementary References

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: Supported by Spanish MICINN grants FIS2009-10061 and FIS2012-33152; CAM research consortium QUITEMAD grant S2009-ESP-1594; European Commission PICC: FP7 2007-2013, grant 249958; integrated project SIQS (grant 600645); UCM-BS grant GICC-910758; the Austrian Science Fund (FWF) through the SFB FoQus (FWF project no. F4002-N16), and the Institut für Quanteninformation GmbH. This research was partially supported by the U.S. Army Research Office through grant W911NF-14-1-0103. This research was partially funded by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through Army Research Office grant W911NF-10-1-0284. All statements of fact, opinion, or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI, or the U.S. Government.
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