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Fault-tolerant quantum computation with few qubits

Rui Chao, Ben W. Reichardt

arXiv:1705.05365v1quant-ph

TL;DR

Large qubit overhead makes fault-tolerance experiments with multiple encoded qubits difficult, especially when individual logical qubits must be addressed within a code block. The paper combines code symmetries, correlated-fault-catching gadgets, and pieceable fault tolerance to implement encoded operations with few ancillas, achieving Clifford computation on the 15-qubit code with two extra qubits and universality with four.

  • Problem

    Large encoding and error-correction overhead, together with inefficient individual addressing in multi-qubit code blocks, limits small-scale fault-tolerance experiments.

  • Method

    The paper combines code symmetries with small gadgets that catch correlated faults and carefully arranged error-correction procedures for fault-tolerant encoded operations.

  • Results

    Two extra qubits support fault-tolerant encoded Clifford operations on seven qubits encoded in 15, while four extra qubits support a universal gate set.

  • Takeaways & Limitations

    The constructions support substantial fault-tolerant quantum calculations in devices with fewer than 20 physical qubits.

Abstract

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Reliable qubits are difficult to engineer, but standard fault-tolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with fault-tolerance schemes with multiple encoded qubits. The 15-qubit Hamming code protects seven encoded qubits to distance three. We give fault-tolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, 17 total. In particular, individual encoded qubits within the code block can be targeted. Fault-tolerant universal computation is possible with four extra qubits, 19 total. The procedures could enable testing more sophisticated protected circuits in small-scale quantum devices. Our main technique is to use gadgets to protect gates against correlated faults. We also take advantage of special code symmetries, and use pieceable fault tolerance.

I. INTRODUCTION

The paper addresses the high qubit overhead and difficult individual addressing that limit fault-tolerance experiments with multiple encoded qubits. It develops low-overhead fault-tolerant operations using code symmetries, correlated-fault-catching gadgets, and carefully arranged error correction.

  • Motivation: Standard fault-tolerance schemes require substantial physical-qubit overhead, making experiments with multiple encoded qubits difficult.Encoding and error-correction overhead can multiply to thousands of physical qubits per logical qubit.
  • Motivation: Multiple-qubit code blocks offer higher rates but make individual encoded qubits complicated and inefficient to address.A prior optimized same-block CNOT method requires a full ancillary code block for temporary logical-qubit transfer.
  • Contributions: The paper introduces lower-overhead fault-tolerant computation on multiple data qubits in distance-two and distance-three codes.For the 15-qubit code, seven qubits are encoded to distance three; two extra qubits support encoded Clifford operations, while four support a universal gate set.
  • Contributions: For [[n, n −2, 2]] codes, two extra qubits suffice for encoded CNOT and Hadamard gates, while four extra qubits suffice for encoded CCZ when n ≥6.The CCZ construction completes a universal gate set.
  • Results: The constructions enable substantial fault-tolerant quantum calculations in devices with fewer than 20 physical qubits.The 15-qubit distance-three construction uses 17 qubits for Clifford operations and 19 for universal computation.
  • Methods: The main gate-protection method replaces two- and three-qubit gates with gadgets that detect correlated faults, while circuit ordering and partial error correction limit fault propagation.Combined CZ gadgets catch the relevant two-qubit correlated faults, but their order matters because incorrect constructions can leave errors undetected.

B. CCZ gadget

The four-ancilla gadget implements a CCZ while constraining single-fault output errors to those correctable around a perfect CCZ. Its fault-tolerance follows from a CSS-code construction using selected qubit subsets and phase-gate products.

  • Four ancilla qubits implement a CCZ on the black data qubits.
  • A single failed gate with trivial measurement results restricts output errors to Paulis arising from a one-qubit fault before or after perfect CCZ.
  • The gadget’s fault-tolerance analysis uses a general CSS-code claim involving products of C(r−1)Z gates over tuples of subsets S_Z^1,...,S_Z^r.
  • For disjoint subsets, the logical action follows by induction using the involution identity for gates G with G^2 = 1.
  • The construction acts as −1 on codewords with all selected encoded bits equal to one and as +1 when at least one selected bit is zero.

III. FAULT-TOLERANT OPERATIONS FOR [[n, n −2, 2]] CODES

For the distance-two [[n, n−2, 2]] code, the paper develops fault-tolerant operations on arbitrary encoded qubits using only two extra qubits, and adds a four-qubit CCZ procedure for universality.

  • Earlier techniques for the same code required at least 2n extra qubits, including separate code blocks for moving logical qubits before an encoded CNOT.
  • Two extra qubits suffice for fault-tolerant encoded CNOT and Hadamard gates on arbitrary logical qubits.
  • For n ≥ 6, four extra qubits suffice for a fault-tolerant encoded CCZ gate, completing a universal gate set.

A. Permutation symmetries and transversal operations

Permutation symmetries and transversal operations preserve the distance-two code while realizing logical permutations, CNOT, Hadamard, and CZ operations. Additional single-qubit encoded gates extend these operations toward the Clifford group.

  • Permutation symmetries: Logical qubit swaps are implemented by physical qubit permutations, which preserve the code space and are fault tolerant when realized by relabeling or moving qubits.
  • Permutation symmetries: The swap (1,2) implements logical CNOTs into encoded qubit 1, while swap (2,n) implements CNOTs in the opposite direction.
  • Transversal operations: Transversal H⊗n followed by swapping qubits 1 and n implements Hadamard on all n−2 encoded qubits.
  • Transversal operations: Transversal G is a valid logical operation, up to a syndrome correction when n = 2 mod 4, and implements logical CZ gates between all encoded qubits.
  • Clifford operations: The available permutations and transversal operations form smaller groups than the full Clifford group, whose subgroup sizes are compared in Table I.
  • Clifford operations: Fault-tolerant single encoded-qubit Hadamard and two-qubit encoded CZ operations generate a large subgroup of the Clifford group.

B. CZ gate

The encoded CZ1,2 construction is derived from a physical implementation that is not fault tolerant, then protected by replacing each CZ with a fault-detecting gadget using at most two ancillas. A fault-tolerant encoded Hadamard can likewise be implemented with two extra qubits.

  • The initial encoded CZ1,2 implementation is not fault tolerant because some two-qubit faults produce undetectable logical errors.For example, an XX failure can become the undetectable X1X2 error.
  • The CZ gadget catches all bad faults in the initial construction, yielding a fault-tolerant logical CZ1,2 implementation with at most two ancilla qubits at a time.The resulting circuit can be simplified by reusing ancillas and merging gadgets for multiple faults.
  • Further simplifications of the fault-tolerant CZ circuit may be possible.
  • A single encoded Hadamard gate can also be implemented fault tolerantly with two extra qubits.The circuit uses measurement-based portions to catch problematic faults, with X measurements expected to return + and Z measurements 0.

D. Four-ancilla CCZ gate

The paper constructs an encoded CCZ for the distance-two code using a round-robin arrangement of physical CCZ gates, then protects the circuit with four ancillas. Single faults are either caught or reduce to detectable errors, establishing fault tolerance.

  • For n ≥6, encoded CCZ on qubits 1, 2, and 3 uses round-robin CCZ gates over {2, n} × {3, n} × {4, n}.
  • The construction contains one Z, three CZ, and four CCZ gates before fault-tolerant gadgets are substituted.
  • Four ancilla qubits suffice after replacing each CZ and CCZ with the corresponding fault-tolerant gadget.
  • Single gate faults are either caught by the gadgets or produce errors detectable at the end, so the procedure is fault tolerant.X or Y faults retain a weight-one X component, while Z faults commute through the controlled gates.
  • The [[15, 7, 3]] Hamming code packs seven logical qubits into 15 physical qubits and has distance three.It is a self-dual CSS code that tolerates less noise than the [[7, 1, 3]] and [[9, 1, 3]] CSS codes.

C. Permutation symmetries

Permutation symmetries of the Hamming code act nontrivially on its encoded qubits and can be converted into logical CZ circuits. For σ3, the resulting logical action is a cyclic pattern of encoded CZ gates on six qubits.

  • Qubit permutations are fault tolerant because they can be implemented by physically moving qubits or relabeling them.
  • The code’s permutation automorphism group has order 20,160 and is isomorphic to A8 and GL(4, 2).
  • The permutations σ1 and σ2 generate the alternating group A7 on the seven logical qubits.They apply the logical permutations (1, 2, 3) and (3, 4, 5, 6, 7), respectively.
  • The logical effect of σ3 is a 24-CNOT circuit rather than a permutation, fixing the first logical qubit and transforming the remaining six cyclically.For j from 2 through 7, each logical Pauli acquires the product of neighboring Paulis with cyclic wraparound.
  • For a self-dual CSS code, CZ gates along the cycles of any code-preserving permutation fix the code space up to Pauli Z corrections.
  • Applying this construction to σ3 produces logical CZ gates following the cycle (2, 3, 4, 5, 6, 7).The physical cycle construction includes cancellations before yielding the logical action.

CZ gates {8, 9} to {10, 11} and {12, 13} to {14, 15}

A physical CZ circuit induces a specified logical operation but is not fault tolerant until its CZ gates are replaced by gadgets. The compiled circuit is verified by showing that all single-fault error possibilities have distinct syndromes.

  • The permutation (6, 7)(8, 10, 9, 11)(12, 14, 13, 15) fixes the code and corresponds to eight physical CZ gates with a stated logical effect.
  • The generated operations reach only half of the six-qubit ⟨CNOT, H⟩ group, rather than arbitrary combinations.
  • Replacing each physical CZ with the Sec. II A gadget yields a compiled circuit using two ancilla qubits and followed by error correction.
  • With trivial gadget measurements, the compiled circuit admits 70 possible errors, all with distinct syndromes and therefore correctable.The check includes one-qubit faults and errors from inequivalent gadget locations.
  • Error correction must combine the X and Z stabilizer syndromes because the code has 28 possible syndromes but only 1 + 15·3 trivial or weight-one errors.
  • Nontrivial gadget measurements localize the fault, leaving error sets whose members have distinct syndromes.One example contains 16 possible errors, all distinguishable by their syndromes; other outcomes were checked computationally.

E. Round-robin CZ circuits to complete the Clifford group

The authors complete the seven-qubit logical Clifford group using a fault-tolerant round-robin construction with syndrome-extracting circuits, flags, and two ancilla qubits.

  • The initially available operations do not generate the full seven-qubit logical Clifford group, motivating the round-robin construction.
  • The construction uses code symmetries and round-robin circuits, replacing vulnerable CZ gates with gadgets and syndrome-extraction procedures.
  • The unprotected circuit is not fault tolerant because some inequivalent errors share a syndrome when all Z measurements return 0.
  • An extended flag continues into the protected circuit to catch X4 or Y4 faults occurring between syndrome extraction and the CZ operations.
  • With two ancilla qubits, the completed procedure implements the encoded CZ operations while using measurement outcomes to select correction or protected-gate branches.

F. Four-qubit fault-tolerant CCZ for universality

The paper extends its gadget-based, pieceable approach to an encoded CCZ gate, using four ancilla qubits and interspersed error correction to obtain fault tolerance.

  • An encoded CCZ is built from a round-robin circuit containing one Z, six CZ, and 21 CCZ gates.
  • Each CZ and CCZ is replaced by a fault-detecting gadget, with X error correction after gadgets and both X and Z correction at the end.
  • The fault-tolerance proof separates triggered gadgets from untriggered gadgets and analyzes the errors produced in each case.
  • For triggered gadgets, possible X and Z errors have distinct syndromes, allowing X errors to be corrected immediately and Z errors at the end.
  • For untriggered gadgets, the remaining errors are reduced to correctable single-qubit components or structured Z errors whose syndromes distinguish them.

V. CONCLUSION

The authors argue that their space-saving fault-tolerance techniques could support both large-scale quantum computing and nearer-term experiments, while noting opportunities for optimization and adaptation.

  • The techniques are presented as useful for large-scale quantum computers and nearer-term fault-tolerance experiments.
  • The authors state that the techniques could be optimized further and adapted to experimental model systems.
  • The methods may also extend to other codes, especially distance-three CSS codes, whereas non-CSS adaptations can be more difficult to make fault tolerant.
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