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arXiv:1612.09444 · Physical Review A

Construction of optimal resources for concatenated quantum protocols

Alexander Pirker, Julius Wallnöfer, Hans J. Briegel, Wolfgang Dür

Published Jun 26, 2017• 6 citations
arXivHTMLPDF

Abstract

We consider the explicit construction of resource states for measurement-based quantum information processing. We concentrate on special-purpose resource states that are capable to perform a certain operation or task, where we consider unitary Clifford circuits as well as non-trace-preserving completely positive maps, more specifically probabilistic operations including Clifford operations and Pauli measurements. We concentrate on $1\ensuremath{\rightarrow}m$ and $m\ensuremath{\rightarrow}1$ operations, i.e., operations that map one input qubit to $m$ output qubits or vice versa. Examples of such operations include encoding and decoding in quantum error correction, entanglement purification, or entanglement swapping. We provide a general framework to construct optimal resource states for complex tasks that are combinations of these elementary building blocks. All resource states only contain input and output qubits, and are hence of minimal size. We obtain a stabilizer description of the resulting resource states, which we also translate into a circuit pattern to experimentally generate these states. In particular, we derive recurrence relations at the level of stabilizers as key analytical tool to generate explicit (graph) descriptions of families of resource states. This allows us to explicitly construct resource states for encoding, decoding, and syndrome readout for concatenated quantum error correction codes, code switchers, multiple rounds of entanglement purification, quantum repeaters, and combinations thereof (such as resource states for entanglement purification of encoded states).

DOI
10.1103/physreva.95.062332

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We consider the explicit construction of resource states for measurement-based quantum information processing. We concentrate on special-purpose resource states that are capable to perform a certain operation or task, where we consider unitary Clifford circuits as well as non-trace-preserving completely positive maps, more specifically probabilistic operations including Clifford operations and Pauli measurements. We concentrate on 1\ensuremath→m1\ensuremath{\rightarrow}m1\ensuremath→m and m\ensuremath→1m\ensuremath{\rightarrow}1m\ensuremath→1 operations, i.e., operations that map one input qubit to mmm output qubits or vice versa. Examples of such operations include encoding and decoding in quantum error correction, entanglement purification, or entanglement swapping. We provide a general framework to construct optimal resource states for complex tasks that are combinations of these elementary building blocks. All resource states only contain input and output qubits, and are hence of minimal size. We obtain a stabilizer description of the resulting resource states, which we also translate into a circuit pattern to experimentally generate these states. In particular, we derive recurrence relations at the level of stabilizers as key analytical tool to generate explicit (graph) descriptions of families of resource states. This allows us to explicitly construct resource states for encoding, decoding, and syndrome readout for concatenated quantum error correction codes, code switchers, multiple rounds of entanglement purification, quantum repeaters, and combinations thereof (such as resource states for entanglement purification of encoded states).
We investigate measurement-based quantum communication with noisy resource states that are generated by entanglement purification. We consider the transmission of encoded information via noisy quantum channels using a measurement-based implementation of encoding, error correction, and decoding. We show that such an approach offers advantages over direct transmission, gate-based error correction, and measurement-based schemes with direct generation of resource states. We analyze the noise structure of resource states generated by entanglement purification and show that a local error model, i.e., noise acting independently on all qubits of the resource state, is a good approximation in general, and provides an exact description for Greenberger-Horne-Zeilinger states. The latter are resources for a measurement-based implementation of error-correction codes for bit-flip or phase-flip errors. This provides an approach to link the recently found very high thresholds for fault-tolerant measurement-based quantum information processing based on local error models for resource states with error thresholds for gate-based computational models.
Quantum technologies rely on high-quality resource states, such as maximally entangled or private states, which are indispensable for quantum communication and cryptography. In practice, however, these states are inevitably degraded by noise. Distillation protocols aim to recover high-resource states from multiple imperfect copies, and while stabilizer-based methods have demonstrated high performance in entanglement purification, they have yet to be established for broader tasks such as secret-key distillation. This work introduces a unified framework for stabilizer-based resource distillation in systems of prime local dimension. By formulating stabilizer routines as quantum channels and deriving closed-form expressions for their output, we enable the application of stabilizer operations to general input states and diverse distillation objectives. We identify key invariances in resource measures, such as coherent and private information, and demonstrate how they can be leveraged to significantly reduce the numerical complexity of channel optimization. To illustrate the framework's versatility, we introduce several protocols: gF-IMAX for general fidelity optimization, and (S)CI-IMAX and (S)PI-IMAX for optimizing (smooth) coherent and private information in both asymptotic and one-shot regimes. Our numerical results confirm that these protocols effectively tailor stabilizer channels to specific operational tasks, establishing them as a robust and flexible tool for quantum resource distillation.
Measurements and feedforward enhance the power of shallow quantum circuits, enabling the deterministic implementation of global unitary operations and the preparation of long-range entangled states. We establish a general correspondence between all such protocols and quantum error-correcting codes: the circuit preceding the measurements acts as an encoder, and unitary feedforward can eliminate post-selection if and only if the measurement projectors are detectable errors on the codespace. This correspondence provides a common framework for state preparation and implementation of global unitaries with measurements and feedforward, turning both into a code-design problem. For stabilizer codes with Pauli measurements, although the resulting operation can be non-Clifford, we show that its nonstabilizerness originates entirely from the encoder. We overcome this restriction using non-Pauli measurements or non-additive codes, constructing protocols that generate long-range nonstabilizerness while requiring only single-qubit unitary corrections.
Protecting quantum information from noise is essential for reliable quantum computation and quantum memories. Quantum state purification addresses this challenge by processing multiple noisy copies of an unknown quantum state to produce an output state of higher fidelity. Here we establish nearly tight upper and lower bounds on the non-Clifford resources required to implement the symmetric-projection purification protocol. For nnn copies of a ddd-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most εεε using O~(n)\widetilde{O}(n)O(n) T gates, where the tilde-OOO notation suppresses polylogarithmic factors in nnn, ddd, and 1/ε1/ε1/ε. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error ε<121−1/nε<\frac{1}{2}\sqrt{1-1/n}ε<21​1−1/n​ requires at least 2n−22n-22n−2 T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.
Quantum LPDC codes provide a substantial reduction in qubit overhead required for fault-tolerant quantum computation compared to surface code, thanks to their high encoding rate. However, operating simultaneously on multiple logical qubits encoded in the same block is more challenging and may slow down logical operations. Prior work addresses this problem by designing complex resource states to perform logical measurements in LDPC codes. Here, we propose an approach that only consumes cat states. Whereas previous work on cat-based measurements focuses on a single logical measurement, we design a protocol for the joint measurement of ℓ\ellℓ commuting logical operators. The key ingredient is the design of a scheduler code determining the measurement sequence and allowing for the decoding of all logical measurement outcomes. Numerical simulations with the LDPC codes Q70 and Q102 of the walking cat architecture show a speed-up of nearly 3×3\times3× over Viterbi measurements for the measurement of ℓ=20\ell=20ℓ=20 commuting logical operators. Combining our fast logical measurements with a new variant of the CliNR partial error correction scheme, we achieve a speed-up of up to 74×74\times74× for random Clifford circuits. Our approach also applies to non-Clifford gates, producing a speed-up of up to 5×5\times5× for Toffoli gates.
In measurement-based quantum computation (MBQC), a special highly entangled state (called a resource state) allows for universal quantum computation driven by single-qubit measurements and postmeasurement corrections. The large number of qubits necessary to construct the resource state constitutes one of the main down sides to MBQC. However, in some instances it is possible to extend the resource state on the fly, meaning that not every qubit must be realized in the devices simultaneously. We consider the question of the minimal number of physical qubits that must be present in a system to directly implement a given measurement pattern. For measurement patterns which have quantum circuit representation as formalized by the notion of flow, with nnn inputs, nnn outputs, and mmm total qubits, we show that only a minimum of n+1n+1n+1 and mmm qubits are required, while the number of required qubits can be as high as $m\ensuremath{-}2$ for measurement patterns which implement a unitary but do not have a quantum circuit representation, as formalized by the notion of generalized flow (gflow). We discuss the implications of removing the Clifford part of a measurement pattern, using well-established transformation rules for Pauli measurements, for the presence of flow versus gflow, and hence the effect on the minimum number of physical qubits required to directly realize the measurement pattern.