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arXiv:2107.03239 · Nature Communications

The two-qubit singlet/triplet measurement is universal for quantum computing given only maximally-mixed initial states

Terry Rudolph, S. Virmani

Published Nov 28, 2023• 3 citations

Abstract

Abstract In order to delineate which minimalistic physical primitives can enable the full power of universal quantum computing, it has been fruitful to consider various measurement based architectures which reduce or eliminate the use of coherent unitary evolution, and also involve operations that are physically natural. In this context previous works had shown that the triplet-singlet measurement of two qubit angular momentum (or equivalently two qubit exchange symmetry) yields the power of quantum computation given access to a few additional different single qubit states or gates. However, Freedman, Hastings and Shokrian-Zini1 recently proposed a remarkable conjecture, called the ‘STP=BQP’ conjecture, which states that the two-qubit singlet/triplet measurement is quantum computationally universal given only an initial ensemble of maximally mixed single qubits. In this work we prove this conjecture. This provides a method for quantum computing that is fully rotationally symmetric (i.e. reference frame independent), using primitives that are physically very-accessible, naturally resilient to certain forms of error, and provably the simplest possible.

DOI: 10.1038/s41467-023-43481-y

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Abstract In order to delineate which minimalistic physical primitives can enable the full power of universal quantum computing, it has been fruitful to consider various measurement based architectures which reduce or eliminate the use of coherent unitary evolution, and also involve operations that are physically natural. In this context previous works had shown that the triplet-singlet measurement of two qubit angular momentum (or equivalently two qubit exchange symmetry) yields the power of quantum computation given access to a few additional different single qubit states or gates. However, Freedman, Hastings and Shokrian-Zini1 recently proposed a remarkable conjecture, called the ‘STP=BQP’ conjecture, which states that the two-qubit singlet/triplet measurement is quantum computationally universal given only an initial ensemble of maximally mixed single qubits. In this work we prove this conjecture. This provides a method for quantum computing that is fully rotationally symmetric (i.e. reference frame independent), using primitives that are physically very-accessible, naturally resilient to certain forms of error, and provably the simplest possible.

We prove that universal quantum computation is possible using only (i) the physically natural measurement on two qubits which distinguishes the singlet from the triplet subspace and (ii) qubits prepared in almost any three different (potentially highly mixed) states. In some sense this measurement is a 'more universal' dynamical element than a universal two-qubit unitary gate, since the latter must be supplemented by measurement. Because of the rotational invariance of the measurement used, our scheme is robust to collective decoherence in a manner very different to previous proposals—in effect it is only ever sensitive to the relational properties of the qubits.

We consider a model of quantum computation using qubits where it is possible to measure whether a given pair are in a singlet (total spin 0 ) or triplet (total spin 1 ) state. The physical motivation is that we can do these measurements in a way that is protected against revealing other information so long as all terms in the Hamiltonian are SU(2) -invariant. We conjecture that this model is equivalent to BQP. Towards this goal, we show: (1) this model is capable of universal quantum computation with polylogarithmic overhead if it is supplemented by single qubit X and Z gates. (2) Without any additional gates, it is at least as powerful as the weak model of "permutational quantum computation" of Jordan [14, 18]. (3) With postselection, the model is equivalent to PostBQP.

Measurement-based quantum computing (MBQC) promises an alternative approach to quantum computation that has natural compatibility with error-correction codes at the cost of a polynomial increase in physical qubits. MBQC implementations have previously focused on photonic systems where two-qubit gates are difficult. On the other hand, semiconductor spin qubit systems offer fast two-qubit gates via the exchange interaction. To explore the benefits of MBQC on spin systems, two hybrid measurement-exchange schemes for full qubit control with semiconductor double-quantum-dot spin qubits are considered. Protocol 1 fully realizes the cluster-state approach to MBQC but requires singlet-triplet qubits in a magnetic field gradient. Protocol 2 implements a direct measurement-based replacement for more traditional gate-based encoded operations, without need for magnetic field gradients. Both protocols demonstrate full single- and two-qubit control through a combination of interqubit and intraqubit exchange and measurement onto the singlet-triplet basis. We show that both schemes suppress individual qubit spin-state leakage errors and offer fast gate times, up to known phase and Pauli corrections.

We first consider various methods for the indirect implementation of unitary gates. We apply these methods to rederive the universality of 4-qubit measurements based on a scheme much simpler than Nielsen's original construction [quant-ph/0108020]. Then, we prove the universality of simple discrete sets of 2-qubit measurements, again using a scheme simplifying the initial construction [quant-ph/0111122]. Finally, we show how to use a single 4-qubit measurement to achieve universal quantum computation, and outline a proof for the universality of almost all maximally entangling 4-qubit measurements.

Joint measurements of two-Pauli observables are powerful tools for both the control and protection of quantum information. By following a simple recipe for measurement choices, single- and two- qubit rotations using two-Pauli parity and single qubit measurements are guaranteed to be unitary while requiring only a single ancilla qubit. This language for measurement based quantum computing is shown to be directly applicable to encoded double quantum dot singlet-triplet spin qubits, by measuring spin-parity between dots from neighboring qubits. Along with exchange interaction, a complete, leakage free, measurement based gate set can be shown, up to a known Pauli correction. Both theoretically exact spin-parity measurements and experimentally demonstrated asymmetric spin-parity measurements are shown to be viable for achieving the proposed measurement based scheme, provided some extra leakage mitigating measurement steps. This method of spin qubit control offers a leakage suppressed, low resource overhead implementation of a measurement-based control that is viable on current spin qubit processor devices.