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arXiv:cond-mat/0412694 · Physical Review B

Decoherence-free states for charge qubits under local nonuniformity

Tetsufumi Tanamoto, Shinobu Fujita

Published Aug 18, 2005• 5 citations
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Abstract

We analyze the robustness of a decoherence-free (DF) subspace and subsystem in charge qubits, when the difference from the collective decoherence measurement condition is large $(\ensuremath{\sim}5%)$ in the long time period, which is applicable for solid-state qubits using as a quantum memory. We solve master equations of up to four charge qubits and a detector as a quantum point contact. We show that robustness of DF states is strongly affected by local nonuniformities. We also discuss the possible two-qubit logical states by exactly solving the master equations.

DOI
10.1103/physrevb.72.085335

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We analyze the robustness of a decoherence-free (DF) subspace and subsystem in charge qubits, when the difference from the collective decoherence measurement condition is large (\ensuremath∼5(\ensuremath{\sim}5%)(\ensuremath∼5 in the long time period, which is applicable for solid-state qubits using as a quantum memory. We solve master equations of up to four charge qubits and a detector as a quantum point contact. We show that robustness of DF states is strongly affected by local nonuniformities. We also discuss the possible two-qubit logical states by exactly solving the master equations.
It was shown recently [D. A. Lidar et al., Phys. Rev. Lett. 81, 2594 (1998)] that within the framework of the semigroup Markovian master equation, decoherence-free (DF) subspaces exist which are stable to first order in time to a perturbation. Here this result is extended to the non-Markovian regime and generalized. In particular, it is shown that within both the semigroup and the non-Markovian operator sum representation, DF subspaces are stable to all orders in time to a symmetry-breaking perturbation. DF subspaces are thus ideal for quantum memory applications. For quantum computation, however, the stability result does not extend beyond the first order. Thus, to perform robust quantum computation in DF subspaces, they must be supplemented with quantum error correcting codes.
One-way quantum computing [1, 2] is an important approach for quantum computation based on a series of one-qubit measurements starting from a cluster state of a qubit array. Cluster states are highly-entangled states involving all qubits and are typically generated from an Ising-like Hamiltonian, starting from an initial product state |Ψ0〉 ≡ |Ψ(t = 0)〉 = Πi=1|+〉i , where |±〉i = (|0〉i±|1〉i)/ √ 2. Here, |0〉i and |1〉i are the two states of the i-th qubit in an N -qubit system. In ref. [3], we showed that cluster states in charge qubits[4–17] can be created by applying a single gate bias pulse, right after preparing the initial product state (one-step generation method), and are more robust against nonuniformities among qubits than decoherence-free (DF) states [18, 19] under a noise environment generated by a quantum point contact (QPC) detector, which is a sensitive detector of electric charge distribution[20, 21]. However, trap sites are often unavoidable in solid-state qubits owing to their small fabrication size and several experiments show that trap states significantly affect electric transport properties of nanoscale devices[22–25]. Here, we model the trap site as the island (discrete energy state) between double QPCs (DQPC) and investigate robustness of cluster states in charge qubits measured by the DQPC detector (Fig. 1). The charge qubits are based on quantum dots (QD), in which the position of the excess charge in a qubit affects the QPC current electrically, resulting in detection of charged state. Owing to this additional external degrees of freedom, this setup is considered to be a harsher and more realistic condition for qubits than that of ref .[3]. Here, we calculate a time-dependent fidelity of four charge qubits with DQPC by solving density matrix (DM) equations.
In this paper, we study the quantum decoherence induced by accumulation of electron tunnelings during the quantum measurement of a charge qubit. The charge qubit is a single electron confined in coupled quantum dots. The measurement of the qubit states is performed using a quantum point contact. A set of master equations for qubit states is derived within a nonequilibrium perturbation to the equilibrium reservoir due to the electron accumulation between the source and drain of the quantum point contact. The quantum decoherence of the qubit states arose from the electron accumulation during the measurement is explored in this framework, and several interesting results on charge qubit decoherence are obtained.
We propose an encoding scheme to fight against ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$-type local collective noise. In our scheme, a logical qubit consists of two charge qubits coupled by a variable capacitance, and different logical qubits are coupled by a common inductance. A logical qubit provides a decoherence-free subspace (DFS) immune to the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$-type collective noise. We show how to realize universal quantum computation in the DFS. Any two logical qubits can be selectively and controllably coupled by adjusting fluxes through the common inductance and superconducting-quantum-interference-device loops, which makes it possible to scale our scheme up to a large number of logical qubits. Finally, we show how to prepare a logical qubit in the DFS.
Quantum computing promises significant speed-up for certain types of computational problems. However, robust implementations of semiconducting qubits must overcome the effects of charge noise that currently limit coherence during gate operations. Here we describe a scheme for protecting solid-state qubits from uniform electric field fluctuations by generalizing the concept of a decoherence-free subspace for spins, and we propose a specific physical implementation: a quadrupole charge qubit formed in a triple quantum dot. The unique design of the quadrupole qubit enables a particularly simple pulse sequence for suppressing the effects of noise during gate operations. Simulations yield gate fidelities 10-1,000 times better than traditional charge qubits, depending on the magnitude of the environmental noise. Our results suggest that any qubit scheme employing Coulomb interactions (for example, encoded spin qubits or two-qubit gates) could benefit from such a quadrupolar design.
We demonstrate experimentally a robust quantum memory using a magnetic-field-independent hyperfine transition in 9Be+ atomic ion qubits at a magnetic field B approximately = 0.01194 T. We observe that the single physical qubit memory coherence time is greater than 10 s, an improvement of approximately 5 orders of magnitude from previous experiments with 9Be+. We also observe long coherence times of decoherence-free subspace logical qubits comprising two entangled physical qubits and discuss the merits of each type of qubit.